Download Decomposition numbers for finite Coxeter groups and generalised

arXiv:0704.0199v3 [math.CO] 17 Nov 2009
DECOMPOSITION NUMBERS FOR FINITE COXETER GROUPS
AND GENERALISED NON-CROSSING PARTITIONS
C. KRATTENTHALER† AND T. W. MÜLLER
Abstract. Given a finite irreducible Coxeter group W , a positive integer d, and
types T1 , T2 , . . . , Td (in the sense of the classification of finite Coxeter groups), we
compute the number of decompositions c = σ1 σ2 · · · σd of a Coxeter element c of W ,
such that σi is a Coxeter element in a subgroup of type Ti in W , i = 1, 2, . . . , d,
and such that the factorisation is “minimal” in the sense that the sum of the ranks
of the Ti ’s, i = 1, 2, . . . , d, equals the rank of W . For the exceptional types, these
decomposition numbers have been computed by the first author in [“Topics in Discrete Mathematics,” M. Klazar et al. (eds.), Springer–Verlag, Berlin, New York, 2006,
pp. 93–126] and [Séminaire Lotharingien Combin. 54 (2006), Article B54l]. The type
An decomposition numbers have been computed by Goulden and Jackson in [Europ. J.
Combin. 13 (1992), 357–365], albeit using a somewhat different language. We explain
how to extract the type Bn decomposition numbers from results of Bóna, Bousquet,
Labelle and Leroux [Adv. Appl. Math. 24 (2000), 22–56] on map enumeration. Our
formula for the type Dn decomposition numbers is new. These results are then used
to determine, for a fixed positive integer l and fixed integers r1 ≤ r2 ≤ · · · ≤ rl , the
number of multi-chains π1 ≤ π2 ≤ · · · ≤ πl in Armstrong’s generalised non-crossing
partitions poset, where the poset rank of πi equals ri , and where the “block structure”
of π1 is prescribed. We demonstrate that this result implies all known enumerative results on ordinary and generalised non-crossing partitions via appropriate summations.
Surprisingly, this result on multi-chain enumeration is new even for the original noncrossing partitions of Kreweras. Moreover, the result allows one to solve the problem
of rank-selected chain enumeration in the type Dn generalised non-crossing partitions
poset, which, in turn, leads to a proof of Armstrong’s F = M Conjecture in type Dn ,
thus completing a computational proof of the F = M Conjecture for all types. It also
allows to address another conjecture of Armstrong on maximal intervals containing a
random multichain in the generalised non-crossing partitions poset.
1. Introduction
The introduction of non-crossing partitions for finite reflection groups (finite Coxeter
groups) by Bessis [8] and Brady and Watt [15] marks the creation of a new, exciting
subject of combinatorial theory, namely the study of these new combinatorial objects
which possess numerous beautiful properties, and seem to relate to several other objects of combinatorics and algebra, most notably to the cluster complex of Fomin and
2000 Mathematics Subject Classification. Primary 05E15; Secondary 05A05 05A10 05A15 05A18
06A07 20F55 33C05.
Key words and phrases. root systems, reflection groups, Coxeter groups, generalised non-crossing
partitions, annular non-crossing partitions, chain enumeration, Möbius function, M -triangle, generalised cluster complex, face numbers, F -triangle, Chu–Vandermonde summation.
†
Research partially supported by the Austrian Science Foundation FWF, grant S9607-N13, in the
framework of the National Research Network “Analytic Combinatorics and Probabilistic Number
Theory”.
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C. KRATTENTHALER AND T. W. MÜLLER
Zelevinsky [21] (cf. [2, 3, 5, 4, 8, 9, 14, 15, 16, 17, 20]). They reduce to the classical
non-crossing partitions of Kreweras [30] for the irreducible reflection groups of type An
(i.e., the symmetric groups), and to Reiner’s [32] type Bn non-crossing partitions for
the irreducible reflections groups of type Bn . (They differ, however, from the type Dn
non-crossing partitions of [32].) The subject has been enriched by Armstrong through
introduction of his generalised non-crossing partitions for reflection groups in [1]. In
the symmetric group case, these reduce to the m-divisible non-crossing partitions of
Edelman [18], while they produce new combinatorial objects already for the reflection
groups of type Bn . Again, these generalised non-crossing partitions possess numerous
beautiful properties, and seem to relate to several other objects of combinatorics and
algebra, most notably to the generalised cluster complex of Fomin and Reading [19] (cf.
[1, 6, 7, 20, 27, 28, 29, 36, 37, 38]).
From a technical point of view, the main subject matter of the present paper is
the computation of the number of certain factorisations of the Coxeter element of a
reflection group. These decomposition numbers, as we shall call them from now on
(see Section 2 for the precise definition), arose in [27, 28], where it was shown that
they play a crucial role in the computation of enumerative invariants of (generalised)
non-crossing partitions. Moreover, in these two papers the decomposition numbers for
the exceptional reflection groups have been computed, and it was pointed out that the
decomposition numbers in type An (i.e., the decomposition numbers for the symmetric
groups) had been earlier computed by Goulden and Jackson in [23]. Here, we explain
how the decomposition numbers in type Bn can be extracted from results of Bóna,
Bousquet, Labelle and Leroux [12] on the enumeration of certain planar maps, and we
find formulae for the decomposition numbers in type Dn , thus completing the project
of computing the decomposition numbers for all the irreducible reflection groups.
The main goal of the present paper, however, is to access the enumerative theory
of the generalised non-crossing partitions of Armstrong via these decomposition numbers. Indeed, one finds numerous enumerative results on ordinary and generalised
non-crossing partitions in the literature (cf. [1, 2, 4, 8, 9, 18, 30, 32, 37]): results on
the total number of (generalised) non-crossing partitions of a given size, of those with a
fixed number of blocks, of those with a given block structure, results on the number of
(multi-)chains of a given length in a given poset of (generalised) non-crossing partitions,
results on rank-selected chain enumeration (that is, results on the number of chains in
which the ranks of the elements of the chains have been fixed), etc. We show that not
only can all these results be rederived from our decomposition numbers, we are also
able to find several new enumerative results. In this regard, the most general type of
result that we find is formulae for the number of (multi-)chains π1 ≤ π2 ≤ · · · ≤ πl−1 in
the poset of non-crossing partitions of type An , Bn , respectively Dn , in which the block
structure of π1 is fixed as well as the ranks of π2 , . . . , πl−1 . Even the corresponding
result in type An , for the non-crossing partitions of Kreweras, is new. Furthermore,
from the result in type Dn , by a suitable summation, we are able to find a formula for
the rank-selected chain enumeration in the poset of generalised non-crossing partitions
of type Dn , thus generalising the earlier formula of Athanasiadis and Reiner [4] for
the rank-selected chain enumeration of “ordinary” non-crossing partitions of type Dn .
In conjunction with the results from [27, 28], this generalisation in turn allows us to
complete a computational case-by-case proof of Armstrong’s “F = M Conjecture” [1,
Conjecture 5.3.2] predicting a surprising relationship between a certain face count in the
DECOMPOSITION NUMBERS FOR FINITE COXETER GROUPS
3
generalised cluster complex of Fomin and Reading and the Möbius function in the poset
of generalised non-crossing partitions of Armstrong. (A case-free proof had been found
earlier by Tzanaki in [38].) Our results allow us also to address another conjecture of
Armstrong [1, Conj. 3.5.13] on maximal intervals containing a random multichain in
the poset of generalised non-crossing partitions. We show that the conjecture is indeed
true for types An and Bn , but that it fails for type Dn (and we suspect that it will also
fail for most of the exceptional types).
We remark that a totally different approach to the enumerative theory of (generalised) non-crossing partitions is proposed in [29]. This approach is, however, completely combinatorial and avoids, in particular, reflection groups. It is, therefore, not
capable of computing our decomposition numbers nor anything else which is intrinsic to
the combinatorics of reflection groups. A similar remark applies to [31, Theorem 4.1],
where a remarkable uniform recurrence is found for rank-selected chain enumeration
in the generalised non-crossing partitions of any type. It could be used, for example,
for verifying our result in Corollary 19 on the rank-selected chain enumeration in the
generalised non-crossing partitions of type Dn , but it is not capable of computing our
decomposition numbers nor of verifying results with restrictions on block structure.
Our paper is organised as follows. In the next section we define the decomposition
numbers for finite reflection groups from [27, 28], the central objects in our paper,
together with a combinatorial variant, which depends on combinatorial realisations of
non-crossing partitions, which we also explain in the same section. This is followed by
an intermediate section in which we collect together some auxiliary results that will
be needed later on. In Section 4, we recall Goulden and Jackson’s formula [23] for
the full rank decomposition numbers of type An , together with the formula from [28,
Theorem 10] that it implies for the decomposition numbers of type An of arbitrary rank.
The purpose of Section 5 is to explain how formulae for the decomposition numbers of
type Bn can be extracted from results of Bóna, Bousquet, Labelle and Leroux in [12].
The type Dn decomposition numbers are computed in Section 6. The approach that
we follow is, essentially, the approach of Goulden and Jackson in [23]: we translate the
counting problem into the problem of enumerating certain maps. This problem is then
solved by a combinatorial decomposition of these maps, translating the decomposition
into a system of equations for corresponding generating functions, and finally solving
this system with the help of the multidimensional Lagrange inversion formula of Good.
Sections 7–11 form the “applications part” of the paper. In the preparatory Section 7,
we recall the definition of the generalised non-crossing partitions of Armstrong, and we
explain the combinatorial realisations of the generalised non-crossing partitions for the
types An , Bn , and Dn from [1] and [29]. The bulk of the applications is contained in
Section 8, where we present three theorems, Theorems 11, 13, and 15, on the number of
factorisations of a Coxeter element of type An , Bn , respectively Dn , with less stringent
restrictions on the factors than for the decomposition numbers. These theorems result
from our formulae for the (combinatorial) decomposition numbers upon appropriate
summations. Subsequently, it is shown that the corresponding formulae imply all known
enumeration results on non-crossing partitions and generalised non-crossing partitions,
plus several new ones, see Corollaries 12, 14, 16–19 and the accompanying remarks.
Section 9 presents the announced computational proof of the F = M (ex-)Conjecture for
type Dn , based on our formula in Corollary 19 for the rank-selected chain enumeration
in the poset of generalised non-crossing partitions of type Dn , while Section 10 addresses
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C. KRATTENTHALER AND T. W. MÜLLER
Conjecture 3.5.13 from [1], showing that it does not hold in general since it fails in type
Dn . In the final Section 11 we point out that the decomposition numbers do not only
allow one to derive enumerative results for the generalised non-crossing partitions of the
classical types, they also provide all the means for doing this for the exceptional types.
For the convenience of the reader, we list the values of the decomposition numbers for
the exceptional types that have been computed in [27, 28] in an appendix.
In concluding the introduction, we want to attract the reader’s attention to the
fact that many of the formulae presented here are very combinatorial in nature (see
Sections 4, 5, 8). This raises the natural question as to whether it is possible to find
combinatorial proofs for them. Indeed, a combinatorial (and, in fact, almost bijective)
proof of the formula of Goulden and Jackson, presented here in Theorem 5, has been
given by Bousquet, Chauve and Schaeffer in [13]. Moreover, most of the proofs for the
known enumeration results on (generalised) non-crossing partitions presented in [1, 2,
4, 18, 32] are combinatorial. On the other hand, to our knowledge so far nobody has
given a combinatorial proof for Theorem 7, the formula for the decomposition numbers
of type Bn , essentially due to Bóna, Bousquet, Labelle and Leroux [12], although we
believe that this should be possible by modifying the ideas from [13]. There are also
other formulae in our paper (see e.g. Corollaries 12 and 14, Eqs. (6.1) and (8.33)) which
seem amenable to combinatorial proofs. However, to find combinatorial proofs for our
type Dn results (cf. in particular Theorem 9.(ii) and Corollaries 16–19) seems rather
hopeless to us.
2. Decomposition numbers for finite Coxeter groups
In this section, we introduce the decomposition numbers from [27, 28], which are
(Coxeter) group-theoretical in nature, plus combinatorial variants for Coxeter groups
of types Bn and Dn , which will be important in combinatorial applications. These
variants depend on the combinatorial realisation of these Coxeter groups, which we
also explain here.
Let Φ be a finite root system of rank n. (We refer the reader to [24] for all terminology
on root systems.) For an element α ∈ Φ, let tα denote the corresponding reflection in
the central hyperplane perpendicular to α. Let W = W (Φ) be the group generated by
these reflections. As is well-known (cf. e.g. [24, Sec. 6.4]), any such reflection group is at
the same time a finite Coxeter group, and all finite Coxeter groups can be realised in this
way. By definition, any element w of W can be represented as a product w = t1 t2 · · · tℓ ,
where the ti ’s are reflections. We call the minimal number of reflections which is needed
for such a product representation the absolute length of w, and we denote it by ℓT (w).
We then define the absolute order on W , denoted by ≤T , via
u ≤T w
if and only if ℓT (w) = ℓT (u) + ℓT (u−1 w).
As is well-known and easy to see, this is equivalent to the statement that every shortest
representation of u by reflections occurs as an initial segment in some shortest product
representation of w by reflections.
Now, for a finite root system Φ of rank n, types T1 , T2 , . . . , Td (in the sense of the
classification of finite Coxeter groups), and a Coxeter element c, the decomposition
number NΦ (T1 , T2 , . . . , Td ) is defined as the number of “minimal” products c1 c2 · · · cd
less than or equal to c in absolute order, “minimal” meaning that ℓT (c1 ) + ℓT (c2 ) +
· · · + ℓT (cd ) = ℓT (c1 c2 · · · cd ), such that, for i = 1, 2, . . . , d, the type of ci as a parabolic
DECOMPOSITION NUMBERS FOR FINITE COXETER GROUPS
5
Coxeter element is Ti . (Here, the term “parabolic Coxeter element” means a Coxeter
element in some parabolic subgroup. The reader should recall that it follows from [8,
Lemma 1.4.3] that any element ci is indeed a Coxeter element in a parabolic subgroup
of W = W (Φ). By definition, the type of ci is the type of this parabolic subgroup. The
reader should also note that, because of the rewriting
−1
−1
c1 c2 · · · cd = ci (c−1
i c1 ci )(ci c2 ci ) · · · (ci ci−1 ci )ci+1 · · · cd ,
(2.1)
any ci in such a minimal product c1 c2 · · · cd ≤T c is itself ≤T c.) It is easy to see that the
decomposition numbers are independent of the choice of the Coxeter element c. (This
follows from the well-known fact that any two Coxeter elements are conjugate to each
other; cf. [24, Sec. 3.16].)
The decomposition numbers satisfy several linear relations between themselves. First
of all, the number NΦ (T1 , T2 , . . . , Td ) is independent of the order of the types T1 , T2 , . . . ,
Td ; that is, we have
NΦ (Tσ(1) , Tσ(2) , . . . , Tσ(d) ) = NΦ (T1 , T2 , . . . , Td )
(2.2)
for every permutation σ of {1, 2, . . . , d}. This is, in fact, a consequence of the rewriting
(2.1). Furthermore, by the definition of these numbers, those of “lower rank” can be
computed from those of “full rank.” To be precise, we have
X
NΦ (T1 , T2 , . . . , Td ) =
NΦ (T1 , T2 , . . . , Td , T ),
(2.3)
T
where the sum is taken over all types T of rank n − rk T1 − rk T2 − · · · − rk Td (with
rk T denoting the rank of the root system Ψ of type T , and n still denoting the rank of
the fixed root system Φ; for later use we record that
ℓT (w0 ) = rk T0
(2.4)
for any parabolic Coxeter element w0 of type T0 ).
The decomposition numbers for the exceptional types have been computed in [27,
28]. For the benefit of the reader, we reproduce these numbers in the appendix. The
decomposition numbers for type An are given in Section 4, the ones for type Bn are
computed in Section 5, while the ones for type Dn are computed in Section 6.
Next we introduce variants of the above decomposition numbers for the types Bn
and Dn , which depend on the combinatorial realisation of the Coxeter groups of these
types.
As is well-known, the reflection group W (An ) can be realised as the symmetric group
Sn+1 on {1, 2, . . . , n + 1}. The reflection groups W (Bn ) and W (Dn ), on the other hand,
can be realised as subgroups of the symmetric group on 2n elements. (See e.g. [11,
Sections 8.1 and 8.2].) Namely, the reflection group W (Bn ) can be realised as the
subgroup of the group of all permutations π of
{1, 2, . . . , n, 1̄, 2̄, . . . , n̄}
satisfying the property
π(ī) = π(i).
(2.5)
¯
(Here, and in what follows, ī is identified with i for all i.) In this realisation, there
is an analogue of the disjoint cycle decomposition of permutations. Namely, every
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C. KRATTENTHALER AND T. W. MÜLLER
π ∈ W (Bn ) can be decomposed as
π = κ1 κ2 · · · κs ,
(2.6)
where, for i = 1, 2, . . . , s, κi is of one of two possible types of “cycles”: a type A cycle,
by which we mean a permutation of the form
((a1 , a2 , . . . , ak )) := (a1 , a2 , . . . , ak ) (a1 , a2 , . . . , ak ),
(2.7)
or a type B cycle, by which we mean a permutation of the form
[a1 , a2 , . . . , ak ] := (a1 , a2 , . . . , ak , a1 , a2 , . . . , ak ),
(2.8)
a1 , a2 , . . . , ak ∈ {1, 2, . . . , n, 1̄, 2̄, . . . , n̄}. (Here we adopt notation from [15].) In both
cases, we call k the length of the “cycle.” The decomposition (2.6) is unique up to a
reordering of the κi ’s.
We call a type A cycle of length k of combinatorial type Ak−1 , while we call a type
B cycle of length k of combinatorial type Bk , k = 1, 2, . . . . The reader should observe
that, when regarded as a parabolic Coxeter element, for k ≥ 2 a type A cycle of length
k has type Ak−1 , while a type B cycle of length k has type Bk . However, a type B
cycle of length 1, that is, a permutation of the form (i, ī), has type A1 when regarded
as a parabolic Coxeter element, while we say that it has combinatorial type B1 . (The
reader should recall that, in the classification of finite Coxeter groups, the type B1 does
not occur, respectively, that sometimes B1 is identified with A1 . Here, when we speak
of “combinatorial type,” then we do distinguish between A1 and B1 . For example,
the “cycles” ((1, 2)) = (1, 2) (1̄, 2̄) or ((1̄, 2)) = (1̄, 2) (1, 2̄) have combinatorial type A1 ,
whereas the cycles [1] = (1, 1̄) or [2] = (2, 2̄) have combinatorial type B1 .)
As Coxeter element for W (Bn ), we choose
c = (1, 2, . . . , n, 1̄, 2̄, . . . , n̄) = [1, 2, . . . , n].
Now, given combinatorial types T1 , T2 , . . . , Td , each of which being a product of Ak ’s
(T1 , T2 , . . . , Td ) is
and Bk ’s, k = 1, 2, . . . , the combinatorial decomposition number NBcomb
n
defined as the number of minimal products c1 c2 · · · cd less than or equal to c in absolute
order, where “minimal” has the same meaning as above, such that for i = 1, 2, . . . , d
the combinatorial type of ci is Ti . Because of (2.1), the combinatorial decomposition
(T1 , T2 , . . . , Td ) satisfy also (2.2) and (2.3).
numbers NBcomb
n
The reflection group W (Dn ) can be realised as the subgroup of the group of all
permutations π of {1, 2, . . . , n, 1̄, 2̄, . . . , n̄} satisfying (2.5) and the property that an even
number of elements from {1, 2, . . . , n} is mapped to an element of negative sign. (Here,
the elements 1, 2, . . . , n are considered to have sign +, while the elements 1̄, 2̄, . . . , n̄ are
considered to have sign −.) Since W (Dn ) is a subgroup of W (Bn ), and since the above
realisation of W (Dn ) is contained as a subset in the realisation of W (Bn ) that we just
described, any π ∈ W (Dn ) can be decomposed as in (2.6), where, for i = 1, 2, . . . , d, κi
is either a type A or a type B cycle. Requiring that π is in the subgroup W (Dn ) of
W (Bn ) is equivalent to requiring that there is an even number of type B cycles in the
decomposition (2.6). Again, the decomposition (2.6) for π ∈ W (Dn ) is unique up to a
reordering of the κi ’s.
As Coxeter element, we choose
c = (1, 2, . . . , n − 1, 1̄, 2̄, . . . , n − 1) (n, n̄) = [1, 2, . . . , n − 1] [n].
DECOMPOSITION NUMBERS FOR FINITE COXETER GROUPS
7
We shall be entirely concerned with elements π of W (Dn ) which are less than or equal
to c. It is not difficult to see (and it is shown in [4, Sec. 3]) that the unique factorisation
of any such element π has either 0 or 2 type B cycles, and in the latter case one of the
type B cycles is [n] = (n, n̄). In this latter case, in abuse of terminology, we call the
product of these two type B cycles, [a1 , a2 , . . . , ak−1 ] [n] say, a “cycle” of combinatorial
type Dk . More generally, we shall say for any product of two disjoint type B cycles of
the form
[a1 , a2 , . . . , ak−1 ] [ak ]
(2.9)
that it is a “cycle” of combinatorial type Dk . The reader should observe that, when
regarded as parabolic Coxeter element, for k ≥ 4 an element of the form (2.9) has type
Dk . However, if k = 3, it has type A3 when regarded as parabolic Coxeter element,
while we say that it has combinatorial type D3 , and, if k = 2, it has type A21 when
regarded as parabolic Coxeter element, while we say that it has combinatorial type
D2 . (The reader should recall that, in the classification of finite Coxeter groups, the
types D3 and D2 do not occur, respectively, that sometimes D3 is identified with A3 ,
D2 being identified with A21 . Here, when we speak of “combinatorial type,” then we do
distinguish between D3 and A3 , and between D2 and A21 .)
Now, given combinatorial types T1 , T2 , . . . , Td , each of which being a product of Ak ’s
(T1 , T2 , . . . , Td ) is
and Dk ’s, k = 1, 2, . . . , the combinatorial decomposition number NBcomb
n
defined as the number of minimal products c1 c2 · · · cd less than or equal to c in absolute
order, where “minimal” has the same meaning as above, such that for i = 1, 2, . . . , d
the combinatorial type of ci is Ti . Because of (2.1), the combinatorial decomposition
(T1 , T2 , . . . , Td ) satisfy also (2.2) and (2.3).
numbers NDcomb
n
3. Auxiliary results
In our computations in the proof of Theorem 9, leading to the determination of the
decomposition numbers of type Dn , we need to apply the Lagrange–Good inversion
formula [22] (see also [26, Sec. 5] and the references cited therein). We recall it here for
the convenience of the reader. In doing so, we use standard multi-index notation. Namely, given a positive integer d, and vectors z = (z1 , z2 , . . . , zd ) and n = (n1 , n2 , . . . , nd ),
we write zn for z1n1 z2n2 · · · zdnd . Furthermore, in abuse of notation, given a formal power
series f in d variables, f (z) stands for f (z1 , z2 , . . . , zd ). Moreover, given d formal power
series f1 , f2 , . . . , fd in d variables, f n (z) is short for
f1n1 (z1 , z2 , . . . , zd )f2n2 (z1 , z2 , . . . , zd ) · · · fdnd (z1 , z2 , . . . , zd ).
Finally, if m = (m1 , m2 , . . . , md ) is another vector, then m + n is short for (m1 +
n1 , m2 + n2 , . . . , md + nd ). Notation such as m − n has to be interpreted in a similar
way.
Theorem 1 (Lagrange–Good inversion). Let d be a positive integer, and let
f1 (z), f2(z), . . . , fd (z) be formal power series in z = (z1 , z2 , . . . , zd ) with the property
that, for all i, fi (z) is of the form zi /ϕi (z) for some formal power series ϕi (z) with
ϕi (0, 0, . . . , 0) 6= 0. Then, if we expand a formal power series g(z) in terms of powers
of the fi (z),
X
g(z) =
γn f n (z),
(3.1)
n
8
C. KRATTENTHALER AND T. W. MÜLLER
the coefficients γn are given by
γn = z−e g(z)f −n−e(z) det
1≤i,j≤d
∂fi
(z) ,
∂zj
where e = (1, 1, . . . , 1), where the sum in (3.1) runs over all d-tuples n of non-negative
integers, and where hzm i h(z) denotes the coefficient of zm in the formal Laurent series
h(z).
Next, we prove a determinant lemma and a corollary, both of which will also be used
in the proof of Theorem 9.
Lemma 2. Let d be a positive integer, and let X1 , X2 , . . . , Xd , Y2 , Y3, . . . , Yd be indeterminates. Then
!
d
d
X
X


Y
Xi −
Yi Y2 Y3 · · · Yd

1 − χ(1 6= j) j , i = 1



i=1
i=2
X1
det 
,
(3.2)
=
Y
1≤i,j≤d

X1 X2 · · · Xd
 1 − χ(i 6= j) i , i ≥ 2

Xi
where χ(S) = 1 if S is true and χ(S) = 0 otherwise.
Proof. By using multilinearity in the rows, we rewrite the determinant on the left-hand
side of (3.2) as
1
X1 − χ(1 6= j)Yj , i = 1
.
det
Xi − χ(i 6= j)Yi , i ≥ 2
X1 X2 · · · Xd 1≤i,j≤d
Next, we subtract the first column from all other columns. As a result, we obtain the
determinant


X1 ,
i=j=1 




 −Yj ,
1
i
=
1
and
j
≥
2
.

det

Xi − Yi , i ≥ 2 and j = 1
X1 X2 · · · Xd 1≤i,j≤d 



χ(i = j)Y ,
i, j ≥ 2
i
Now we add rows 2, 3, . . . , d to the first row. After that, our determinant
becomes
lower
Pd
Pd
triangular, with the entry in the first row and column equal to i=1 Xi − i=2 Yi , and
the diagonal entry in row i, i ≥ 2, equal to Yi . Hence, we obtain the claimed result. Corollary 3. Let d and r be positive integers, 1 ≤ r ≤ d, and let X1 , X2 , . . . , Xd , Y
and Z be indeterminates. Then, with notation as in Lemma 2,
1 − χ(r 6= j) XZr , i = r
det
1 − χ(i 6= j) XY i , i 6= r
1≤i,j≤d
P
Y d−2 Z di=1 Xi + (Y − Z)Xr − (d − 1)Y Z
. (3.3)
=
X1 X2 · · · Xd
Proof. We write the diagonal entry in the r-th row of the determinant in (3.3) as
1=
Xr + Y − Z Y − Z
−
,
Xr
Xr
DECOMPOSITION NUMBERS FOR FINITE COXETER GROUPS
9
and then use linearity of the determinant in the r-th row to decompose the determinant
as
Xr + Y − Z
Y −Z
D1 −
D2 ,
Xr
Xr
where D1 is the determinant in (3.2) with Xr replaced by Xr + Y − Z, and with Yi = Y
for all i, and where D2 is the determinant in (3.2) with d replaced by d − 1, with Yi = Y
for all i, and with Xi replaced by Xi−1 for i = r + 1, r + 2, . . . , d. Hence, using Lemma 2,
we deduce that the determinant in (3.3) is equal to
Pd
Pd
d−2
X
+
Y
−
Z
−
(d
−
1)Y
Y d−1
X
−
X
−
(d
−
2)Y
(Y
−
Z)Y
i
i
r
i=1
i=1
−
.
X1 X2 · · · Xd
X 1 X 2 · · · Xd
Little simplification then leads to (3.3).
We end this section with a summation lemma, which we shall need in Sections 5 and 6
in order to compute the Bn , respectively Dn , decomposition numbers of arbitrary rank
from those of full rank, and in Section 8 to derive enumerative results for (generalised)
non-crossing partitions from our formulae for the decomposition numbers.
Lemma 4. Let M and r be non-negative integers. Then
X
M +r−1
M
,
=
r
m
,
m
,
.
.
.
,
m
1
2
r
m +2m +···+rm =r
1
2
(3.4)
r
where the multinomial coefficient is defined by
M!
M
.
=
m1 ! m2 ! · · · mr ! (M − m1 − m2 − · · · − mr )!
m1 , m2 , . . . , mr
Proof. The identity results directly by comparing coefficients of z r on both sides of the
identity
(1 + z + z 2 + z 3 + · · · )M = (1 − z)−M .
4. Decomposition numbers for type A
As was already pointed out in [28, Sec. 10], the decomposition numbers for type An
have already been computed by Goulden and Jackson in [23, Theorem 3.2], albeit using
a somewhat different language. (The condition on the sum l(α1 ) + l(α2 ) + · · · + l(αm )
is misstated throughout the latter paper. It should be replaced by l(α1 ) + l(α2 ) + · · · +
l(αm ) = (m − 1)n + 1.) In our terminology, their result reads as follows.
Theorem 5. Let T1 , T2 , . . . , Td be types with rk T1 + rk T2 + · · · + rk Td = n, where
(i)
m
(i)
(i)
m
n
Ti = A1 1 ∗ A2 2 ∗ · · · ∗ Am
n ,
i = 1, 2, . . . , d.
Then
d−1
NAn (T1 , T2 , . . . , Td ) = (n + 1)
d
Y
i=1
n − rk Ti + 1
1
(i)
(i) ,
n − rk Ti + 1 m(i)
1 , m2 , . . . , mn
where the multinomial coefficient is defined as in Lemma 4.
(4.1)
10
C. KRATTENTHALER AND T. W. MÜLLER
Here we have used Stembridge’s [35] notation for the decomposition of types into a
product of irreducibles; for example, the equation T = A32 ∗ A5 means that the root
system of type T decomposes into the orthogonal product of 3 copies of root systems
of type A2 and one copy of the root system of type A5 .
It was shown in [28, Theorem 10] that, upon applying the summation formula in
Lemma 4 to the result in Theorem 5 in a suitable manner, one obtains a compact
formula for all type An decomposition numbers.
Theorem 6. Let the types T1 , T2 , . . . , Td be given, where
(i)
(i)
m
(i)
m
n
Ti = A1 1 ∗ A2 2 ∗ · · · ∗ Am
n ,
i = 1, 2, . . . , d.
Then
d−1
NAn (T1 , T2 , . . . , Td ) = (n + 1)
n+1
rk T1 + rk T2 + · · · + rk Td + 1
d
Y
n − rk Ti + 1
1
×
(i)
(i) , (4.2)
n − rk Ti + 1 m(i)
1 , m2 , . . . , mn
i=1
where the multinomial coefficient is defined as in Lemma 4. All other decomposition
numbers NAn (T1 , T2 , . . . , Td ) are zero.
5. Decomposition numbers for type B
In this section we compute the decomposition numbers in type Bn . We show that one
can extract the corresponding formulae from results of Bóna, Bousquet, Labelle and
Leroux [12] on the enumeration of certain planar maps, which they call m-ary cacti.
While reading the statement of the theorem, the reader should recall from Section 2
the distinction between group-theoretic and combinatorial decomposition numbers.
Theorem 7. (i) If T1 , T2 , . . . , Td are types with rk T1 + rk T2 + · · · + rk Td = n, where
(i)
m
(i)
(i)
m
n
Ti = A1 1 ∗ A2 2 ∗ · · · ∗ Am
n ,
i = 1, 2, . . . , j − 1, j + 1, . . . , d,
and
(j)
m
(j)
(j)
m
n
Tj = Bα ∗ A1 1 ∗ A2 2 ∗ · · · ∗ Am
,
n
for some α ≥ 1, then
(T1 , T2 , . . . , Td )
NBcomb
n
d−1
=n
n − rk Tj
(j)
(j)
(j)
m1 , m2 , . . . , mn
Y
d
n − rk Ti
1
(i)
(i) ,
n − rk Ti m(i)
1 , m2 , . . . , mn
i=1
i6=j
(5.1)
where the multinomial coefficient is defined as in Lemma 4. For α ≥ 2, the number
NBn (T1 , T2 , . . . , Td ) is given by the same formula.
(ii) If T1 , T2 , . . . , Td are types with rk T1 + rk T2 + · · · + rk Td = n, where
(i)
m
(i)
m
(i)
n
Ti = A1 1 ∗ A2 2 ∗ · · · ∗ Am
n ,
i = 1, 2, . . . , d,
DECOMPOSITION NUMBERS FOR FINITE COXETER GROUPS
11
then
! X
d
(j)
n
−
rk
T
1
m1 (n − rk Tj )
i
d−1
NBn (T1 , T2 , . . . , Td ) = n
,
(i)
(i)
(j)
n − rk Ti m(i)
m0 + 1
1 , m2 , . . . , mn
i=1
j=1
(5.2)
Pn
(j)
(j)
where m0 = n − rk Tj − s=1 ms .
(T1 , T2 , . . . , Td )
(iii) All other decomposition numbers NBn (T1 , T2 , . . . , Td ) and NBcomb
n
with rk T1 + rk T2 + · · · + rk Td = n are zero.
d
Y
Proof. Determining the decomposition numbers
NBn (T1 , T2 , . . . , Td ) = NBn (Td , . . . , T2 , T1 )
(recall (2.2)), respectively
(Td , . . . , T2 , T1 ),
(T1 , T2 , . . . , Td ) = NBcomb
NBcomb
n
n
amounts to counting all possible factorisations
[1, 2, . . . , n] = σd · · · σ2 σ1 ,
(5.3)
where σi has type Ti as a parabolic Coxeter element, respectively has combinatorial
type Ti . The reader should observe that the factorisation (5.3) is minimal, in the sense
that
n = ℓT [1, 2, . . . , n] = ℓT (σ1 ) + ℓT (σ2 ) + · · · + ℓT (σd ),
since ℓT (σi ) = rk Ti , and since, by our assumption, the sum of the ranks of the Ti ’s
equals n. A further observation is that, in a factorisation (5.3), there must be at least
one factor σi which contains a type B cycle in its (type B) disjoint cycle decomposition,
because the sign of [1, 2, . . . , n] as an element of the group S2n of all permutations of
{1, 2, . . . , n, 1̄, 2̄, . . . , n̄} is −1, while the sign of any type A cycle is +1.
We first prove Claim (iii). Let us assume, by contradiction, that there is a minimal
decomposition (5.3) in which, altogether, we find at least two type B cycles in the (type
B) disjoint cycle decompositions of the σi ’s. In that case, (5.3) has the form
[1, 2, . . . , n] = u1 κ1 u2 κ2 u3 ,
(5.4)
where κ1 and κ2 are two type B cycles, and u1 , u2, u3 are the factors in between.
Moreover, the factorisation (5.4) is minimal, meaning that
n = ℓT (u1 ) + ℓT (κ1 ) + ℓT (u2 ) + ℓT (κ2 ) + ℓT (u3 ).
(5.5)
We may rewrite (5.4) as
−1
−1
[1, 2, . . . , n] = κ1 κ2 (κ−1
2 κ1 u1 κ1 κ2 )(κ2 u2 κ2 )u3 ,
−1
−1
′
or, setting u′1 = κ−1
2 κ1 u1 κ1 κ2 and u2 = κ2 u2 κ2 , as
[1, 2, . . . , n] = κ1 κ2 u′1 u′2 u3 .
(5.6)
This factorisation is still minimal since u′1 is conjugate to u1 and u′2 is conjugate to
u2 . At this point, we observe that κ1 must be a cycle of the form (2.8) with a1 <
a2 < · · · < ak < a1 < a2 < · · · < ak in the order 1 < 2 < · · · < n < 1̄ < 2̄ < · · · < n̄,
because otherwise κ1 6≤T [1, 2, . . . , n], which would contradict (5.6). A similar argument
12
C. KRATTENTHALER AND T. W. MÜLLER
3
4̄
1
23
1
1
7̄
5̄
3
1
10
3
3̄
2
1
3
2̄
3
3
1
2 3
1 3
8
1̄
6̄
2
3
6
1
9
7
2
2
1
3
3
2
1 3
1
9̄
3
1
10
8̄
3
2
3
1
3
2
2
1
3
5
4
1
3
Figure 1. The 3-cactus corresponding to the factorisation (5.7)
applies to κ2 . Now, if κ1 and κ2 are not disjoint, then it is easy to see that ℓT (κ1 κ2 ) <
ℓT (κ1 ) + ℓT (κ2 ), whence
n = ℓT ([1, 2, . . . , n])
= ℓT (κ1 κ2 u′1 u′2 u3 )
≤ ℓT (κ1 κ2 ) + ℓT (u′1) + ℓT (u′2) + ℓT (u3 )
≤ ℓT (κ1 κ2 ) + ℓT (u1) + ℓT (u2) + ℓT (u3 )
< ℓT (κ1 ) + ℓT (κ2 ) + ℓT (u1 ) + ℓT (u2 ) + ℓT (u3 ),
a contradiction to (5.5). If, on the other hand, κ1 and κ2 are disjoint, then we can find
i, j ∈ {1, 2, . . . , n, 1̄, 2̄, . . . , n̄}, such that i < j < κ1 (i) < κ2 (j) (in the above order of
{1, 2, . . . , n, 1̄, 2̄, . . . , n̄}). In other words, if we represent κ1 and κ2 in the obvious way
in a cyclic diagram (cf. [32, Sec. 2]), then they cross each other. However, in that case
we have
κ1 κ2 6≤T [1, 2, . . . , n],
contradicting the fact that (5.6) is a minimal factorisation. (This is one of the consequences of Biane’s group-theoretic characterisation [10, Theorem 1] of non-crossing
partitions.)
DECOMPOSITION NUMBERS FOR FINITE COXETER GROUPS
13
We turn now to Claims (i) and (ii). In what follows, we shall show that the formulae
(5.1) and (5.2) follow from results of Bóna, Bousquet, Labelle and Leroux [12] on the
enumeration of m-ary cacti with a rotational symmetry. In order to explain this, we
must first define a bijection between minimal factorisations (5.3) and certain planar
maps. By a map, we mean a connected graph embedded in the plane such that edges
do not intersect except in vertices. The maps which are of relevance here are maps
in which faces different from the outer face intersect only in vertices, and are coloured
with colours from {1, 2, . . . , d}. Such maps will be referred to as d-cacti from now on.1
Examples of 3-cacti can be found in Figures 1 and 2. In the figures, the faces different
from the outer face are the shaded ones. Their colours are indicated by the numbers 1,
2, respectively 3, placed in the centre of the faces. Figure 1 shows a 3-cactus in which
the vertices are labelled, while Figure 2 shows one in which the vertices are not labelled.
(That one of the vertices in Figure 2 is marked by a bold dot should be ignored for the
moment.)
In what follows, we need the concept of the rotator around a vertex v in a dcactus, which, by definition, is the cyclic list of colours of faces encountered in a
clockwise journey around v. If, while travelling around v, we encounter the colours
b1 , b2 , . . . , bk , in this order, then we will write (b1 , b2 , . . . , bk )O for the rotator, meaning
that (b1 , b2 , . . . , bk )O = (b2 , . . . , bk , b1 )O , etc. For example, the rotator of all the vertices
in the map in Figure 1 is (1, 2, 3)O .
We illustrate the bijection between minimal factorisations (5.3) and d-cacti with an
example. Take n = 10 and d = 3, and consider the factorisation
[1, 2, . . . , 10] = σ3 σ2 σ1 ,
(5.7)
where σ3 = ((7, 8)), σ2 = [2, 6, 8] ((1, 9̄, 10)) ((4, 5)), and σ1 = ((1, 8̄)) ((2, 3, 5)). For
each cycle (a1 , a2 , . . . , ak ) (sic!) of σi , we create a k-gon coloured i, and label its vertices
a1 , a2 , . . . , ak in clockwise order. (The warning “sic!” is there to avoid misunderstandings: for each type A “cycle” ((b1 , b2 , . . . , bk )) we create two k-gons, the vertices of one
being labelled b1 , b2 , . . . , bk , and the vertices of the other being labelled b1 , b2 , . . . , bk ,
while for each type B “cycle” [b1 , b2 , . . . , bk ] we create one 2k-gon with vertices labelled
b1 , b2 , . . . , bk , b1 , b2 , . . . , bk .) We glue these polygons into a d-cactus, the faces of which
are these polygons plus the outer face, by identifying equally labelled vertices such that
the rotator of each vertex is (1, 2, . . . , d). Figure 1 shows the outcome of this procedure
for the factorisation (5.7).
The fact that the result of the procedure can be realised as a d-cactus follows from
Euler’s formula. Namely, the number of faces corresponding to the polygons is 1 +
P P
(i)
2 di=1 nk=0 mk (the 1 coming from the polygon corresponding to the type B cycle),
P P
(i)
the number of edges is 2α + 2 di=1 nk=0 mk (k + 1), and the number of vertices is 2n.
Hence, if we include the outer face, the number of vertices minus the number of edges
1We
warn the reader that our terminology deviates from the one in [12, 23]. We follow loosely the
conventions in [25]. To be precise, our d-cacti in which the rotator around every vertex is (1, 2, . . . , d)O
are dual to the coloured d-cacti in [23], respectively d-ary cacti in [12], in the following sense: one is
obtained from the other by “interchanging” the roles of vertices and faces, that is, given a d-cactus in
our sense, one obtains a d-cactus in the sense of Goulden and Jackson by shrinking faces to vertices
and blowing up vertices of degree δ to faces with δ vertices, keeping the incidence relations between
faces and vertices. Another minor difference is that colours are arranged in counter-clockwise order in
[12, 23], while we arrange colours in clockwise order.
14
C. KRATTENTHALER AND T. W. MÜLLER
3
23
1
3
1
3
2
1
1
3
3
2
1 3
1
3
2
3
3
1
2 3
1 3
2
2
2
3
1
3
2
3
1
1
1
1
3
1
3
3
Figure 2. A rotation-symmetric 3-cactus with a marked vertex
plus the number of faces is
2n − 2α − 2
d X
n
X
i=1 k=0
(i)
mk (k + 1) + 2
d X
n
X
(i)
mk + 2
i=1 k=0
= 2n + 2 − 2α − 2
d X
n
X
(i)
k · mk
i=1 k=0
= 2n + 2 − 2 rk T1 − 2 rk T2 − · · · − 2 rk Td
= 2,
(5.8)
according to our assumption concerning the sum of the ranks of the types Ti .
We may further simplify this geometric representation of a minimal factorisation
(5.3) by deleting all vertex labels and marking the vertex which had label 1. If this
simplification is applied to the 3-cactus in Figure 1, we obtain the 3-cactus in Figure 2.
Indeed, the knowledge of which vertex carries label 1 allows us to reconstruct all other
vertex labels as follows: starting from the vertex labelled 1, we travel clockwise along
the boundary of the face coloured 1 until we reach the next vertex (that is, we traverse
only a single edge); from there, we travel clockwise along the boundary of the face
coloured 2 until we reach the next vertex; etc., until we have travelled along an edge
DECOMPOSITION NUMBERS FOR FINITE COXETER GROUPS
15
bounding a face of colour d. The vertex that we have reached must carry label 2. Etc.
Clearly, if drawn appropriately into the plane, a d-cactus resulting from an application
of the above procedure to a minimal factorisation (5.3) is symmetric with respect to
a rotation by 180◦, the centre of the rotation being the centre of the regular 2α-gon
corresponding to the unique type B cycle of σj ; cf. Figure 2. In what follows, we shall
abbreviate this property as rotation-symmetric.
In summary, under the assumptions of Claim (i), the decomposition number
comb
NBn (T1 , T2 , . . . , Td ), respectively, if α ≥ 2, the decomposition number NBn (T1 , T2 , . . . ,
Td ) also, equals the number of all rotation-symmetric d-cacti on 2n vertices in which
(i)
one vertex is marked and all vertices have rotator (1, 2, . . . , d)O , with exactly mk pairs
of faces of colour i having k + 1 vertices, arranged symmetrically around a central face
of colour j with 2α vertices.
Aside from the marking of one vertex, equivalent objects are counted in [12, Theorem 25]. In our language, modulo the “dualisation” described in Footnote 1, and upon
replacing m by d, the objects which are counted in the cited theorem are d-cacti in
which all vertices have rotator (1, 2, . . . , d)O , and which are invariant under a rotation
(not necessarily by 180◦). To be precise, from the proof of [12, (81)] (not given in full
detail in [12]) it can be extracted that the number of d-cacti on 2n vertices, in which all
vertices have rotator (1, 2, . . . , d)O , which are invariant under a rotation by (360/s)◦, s
(i)
being maximal with this property, and which have exactly 2mk faces of colour i having
k + 1 vertices arranged around a central face of colour j with 2α vertices, equals
X
2(n − rk Tj )/t
d−2
′
(2n) s
µ(t/s)
(j)
(j)
(j)
2m1 /t, 2m2 /t, . . . , 2mn /t
t
d
Y
2(n − rk Ti )/t
1
·
, (5.9)
(i)
(i)
(i)
2(n
−
rk
T
)
2m
/t,
2m
/t,
.
.
.
,
2m
/t
i
n
1
2
i=1
i6=j
(i)
where the sum extends over all t with s | t, t | 2α, and t | 2mk for all i = 1, 2, . . . , d and
k = 1, 2, . . . , n. Here, µ(·) is the Möbius function from number theory.2 In presenting
the formula in the above form, we have also used the observation that, for all i (including
i = j !), the number of type A cycles of σi is n − rk Ti .
As we said above, the d-cacti that we want to enumerate have one marked vertex,
whereas the d-cacti counted by (5.9) have no marked vertex. However, given a d-cactus
counted by (5.9), we have exactly 2n/s inequivalent ways of marking a vertex. Hence,
recalling that the d-cacti that we want to count are invariant under a rotation by 180◦ ,
we must multiply the expression (5.9) by 2n/s, and then sum the result over all even
s. Since, by definition of the Möbius function, we have
(
X
X
1 if 2t = 1,
′
µ(t/s) =
µ(t/2s ) =
0 otherwise,
′ t
2|s|t
s |2
the result of this summation is exactly the right-hand side of (5.1).
2Formula
(81) in [12] does not distinguish the colour or the size of the central face (that is, in the
language of [12]: the colour or the degree of the central vertex), therefore it is in fact a sum over all
possible colours and sizes, represented there by the summations over i and h, respectively.
16
C. KRATTENTHALER AND T. W. MÜLLER
Finally, we prove Claim (ii). From what we already know, in a minimal factorisation
(5.3) exactly one of the factors on the right-hand side must contain a type B cycle of
length 1 in its (type B) disjoint cycle decomposition, σj say. As a parabolic Coxeter
element, a type B cycle of length 1 has type A1 . Since all considerations in the proof
of Claim (i) are also valid for α = 1, we may use Formula (5.1) with α = 1, and with
(j)
(j)
m1 replaced by m1 − 1, to count the number of these factorisations, to obtain
d−1
n
n − rk Tj
(j)
(j)
(j)
m1 − 1, m2 , . . . , mn
n − rk Ti
1
(i)
(i) .
n − rk Ti m(i)
1 , m2 , . . . , mn
i=1
Y
d
i6=j
This has to be summed over j = 1, 2, . . . , d. The result is exactly (5.2).
The proof of the theorem is now complete.
Combining the previous theorem with the summation formula of Lemma 4, we can
now derive compact formulae for all type Bn decomposition numbers.
Theorem 8. (i) Let the types T1 , T2 , . . . , Td be given, where
(i)
m
(i)
(i)
m
n
Ti = A1 1 ∗ A2 2 ∗ · · · ∗ Am
n ,
i = 1, 2, . . . , j − 1, j + 1, . . . , d,
and
(j)
m
(j)
m
(j)
n
Tj = Bα ∗ A1 1 ∗ A2 2 ∗ · · · ∗ Am
,
n
for some α ≥ 1. Then
n
=n
rk T1 + rk T2 + · · · + rk Td
Y
d
n − rk Tj
n − rk Ti
1
×
(i)
(i)
(i) , (5.10)
(j)
(j)
(j)
m1 , m2 , . . . , mn i=1 n − rk Ti m1 , m2 , . . . , mn
NBcomb
(T1 , T2 , . . . , Td )
n
d−1
i6=j
where the multinomial coefficient is defined as in Lemma 4. For α ≥ 2, the number
NBn (T1 , T2 , . . . , Td ) is given by the same formula.
(ii) Let the types T1 , T2 , . . . , Td be given, where
(i)
m
(i)
m
(i)
n
Ti = A1 1 ∗ A2 2 ∗ · · · ∗ Am
n ,
i = 1, 2, . . . , d.
Then
NBcomb
(T1 , T2 , . . . , Td )
n
!
Y
d
n
−
rk
T
1
n
−
1
i
, (5.11)
= nd
(i)
(i)
(i)
n
−
rk
T
rk T1 + rk T2 + · · · + rk Td
m
,
m
,
.
.
.
,
m
n
i
1
2
i=1
DECOMPOSITION NUMBERS FOR FINITE COXETER GROUPS
17
whereas
NBn (T1 , T2 , . . . , Td )
Y
!
d
n
−
rk
T
n
1
i
= nd−1
(i)
(i)
rk T1 + rk T2 + · · · + rk Td
n − rk Ti m(i)
1 , m2 , . . . , mn
i=1
!
d
(j)
X
m1 (n − rk Tj )
× n − rk T1 − rk T2 − · · · − rk Td +
, (5.12)
(j)
m
+
1
0
j=1
P
(j)
(j)
with m0 = n − rk Tj − ns=1 ms .
(T1 , T2 , . . . , Td )
(iii) All other decomposition numbers NBn (T1 , T2 , . . . , Td ) and NBcomb
n
are zero.
Proof. If we write r for n − rk T1 − rk T2 − · · · − rk Td , then for Φ = Bn the relation (2.3)
becomes
X
(5.13)
NBn (T1 , T2 , . . . , Td , T ),
NBn (T1 , T2 , . . . , Td ) =
T :rk T =r
comb
the same relation holding for NBn in place of NBn .
m2
mn
1
order to prove (5.10), we let T = Am
1 ∗ A2 ∗ · · · ∗ An
with
In
obtain
NBcomb
(T1 , T2 , . . . , Td )
n
and use (5.1) in (5.13), to
n−r
1
n
=
n − r m1 , m2 , . . . , mn
m1 +2m2 +···+nmn =r
Y
d
n − rk Tj
n − rk Ti
1
·
(i)
(i)
(i) .
(j)
(j)
(j)
m1 , m2 , . . . , mn i=1 n − rk Ti m1 , m2 , . . . , mn
X
d
i6=j
If we use (3.4) with M = n − r, we arrive at our claim after little simplification.
m2
mn
1
In order to prove (5.11), we let T = Bα ∗ Am
in (5.13). The
1 ∗ A2 ∗ · · · ∗ An
important point to be observed here is that, in contrast to the previous argument, in
the present case T must have a factor Bα . Subsequently, use of (5.1) in (5.13) yields
n
X
X
n−r
d
comb
n
NBn (T1 , T2 , . . . , Td ) =
m1 , m2 , . . . , mn
α=1 m1 +2m2 +···+nmn =r−α
d
Y
n − rk Ti
1
·
(i)
(i) . (5.14)
n − rk Ti m(i)
1 , m2 , . . . , mn
i=1
Now we use (3.4) with r replaced by r − α and M = n − r, and subsequently the
elementary summation formula
X
n n X
n−1
n−1
n−α−1
n−α−1
.
(5.15)
=
=
=
r−1
n−r
n−r−1
r−α
α=1
α=1
Then, after little rewriting, we arrive at our claim.
To establish (5.12), we must recall that the group-theoretic type A1 does not distinguish between a type A cycle ((i, j)) = (i, j) (ī, j̄) and a type B cycle [i] = (i, ī). Hence,
to obtain NBn (T1 , T2 , . . . , Td ) in the case that no Ti contains a Bα for α ≥ 2, we must
18
C. KRATTENTHALER AND T. W. MÜLLER
(j)
(j)
add the expression (5.11) and the expressions (5.10) with m1 replaced by m1 − 1 over
j = 1, 2, . . . , d. As is not difficult to see, this sum is indeed equal to (5.12).
6. Decomposition numbers for type D
In this section we compute the decomposition numbers for the type Dn . Theorem 9
gives the formulae for the full rank decomposition numbers, while Theorem 10 presents
the implied formulae for the decomposition numbers of arbitrary rank. To our knowledge, these are new results, which did not appear earlier in the literature on map
enumeration or on the connection coefficients in the symmetric group or other Coxeter
groups. Nevertheless, the proof of Theorem 9 is entirely in the spirit of the fundamental
paper [23], in that the problem of counting factorisations is translated into a problem of
map enumeration, which is then solved by a generating function approach that requires
the use of the Lagrange–Good formula for coefficient extraction.
We begin with the result concerning the full rank decomposition numbers in type
Dn . While reading the statement of the theorem below, the reader should again recall
from Section 2 the distinction between group-theoretic and combinatorial decomposition
numbers.
Theorem 9. (i) If T1 , T2 , . . . , Td are types with rk T1 + rk T2 + · · · + rk Td = n, where
(i)
m
(i)
(i)
m
n
Ti = A1 1 ∗ A2 2 ∗ · · · ∗ Am
n ,
i = 1, 2, . . . , j − 1, j + 1, . . . , d,
and
(j)
(j)
m
(j)
m
n
Tj = Dα ∗ A1 1 ∗ A2 2 ∗ · · · ∗ Am
,
n
for some α ≥ 2, then
(T1 , T2 , . . . , Td )
NDcomb
n
d−1
= (n − 1)
n − rk Tj
(j)
(j)
(j)
m1 , m2 , . . . , mn
d
Y
n − rk Ti − 1
1
×
(i)
(i) , (6.1)
n − rk Ti − 1 m(i)
1 , m2 , . . . , mn
i=1
i6=j
where the multinomial coefficient is defined as in Lemma 4. For α ≥ 4, the number
NDn (T1 , T2 , . . . , Td ) is given by the same formula.
(ii) If T1 , T2 , . . . , Td are types with rk T1 + rk T2 + · · · + rk Td = n, where
(i)
m
(i)
m
(i)
n
Ti = A1 1 ∗ A2 2 ∗ · · · ∗ Am
n ,
i = 1, 2, . . . , d,
DECOMPOSITION NUMBERS FOR FINITE COXETER GROUPS
19
then
(T1 , T2 , . . . , Td )
NDcomb
n

Y
d d
 X
n − rk Tj
n − rk Ti − 1
1
(i)
(i)
(i)
(j)
(j)
(j)
2
m1 , m2 , . . . , mn i=1 n − rk Ti − 1 m1 , m2 , . . . , mn
j=1
d−1 
= (n − 1)
i6=j
d
Y
−2(d − 1)(n − 1)


n − rk Ti − 1
1

(i)
(i)
(i)  , (6.2)
n
−
rk
T
−
1
m
,
m
,
.
.
.
,
m
n
i
1
2
i=1
while
NDn (T1 , T2 , . . . , Td )

! d
d
X
Y

n − rk Tj
n
−
rk
T
−
1
1
i
d−1 
= (n − 1) 
2
(i)
(i)
(j)
(j)
(j)
n − rk Ti − 1 m(i)
m1 , m2 , . . . , mn
1 , m2 , . . . , mn
j=1
i=1
i6=j
+
n − rk Tj
(j)
(j)
(j)
(j)
(j)
m1 , m2 , m3 − 1, m4 , . . . , mn
+
n − rk Tj
(j)
(j)
(j)
m1 − 2, m2 , . . . , mn
!


n − rk Ti − 1
1

(i)
(i)
(i)  . (6.3)
n
−
rk
T
−
1
m
,
m
,
.
.
.
,
m
n
i
1
2
i=1
d
Y
−2(d − 1)(n − 1)
(T1 , T2 , . . . , Td )
(iii) All other decomposition numbers NDn (T1 , T2 , . . . , Td ) and NDcomb
n
with rk T1 + rk T2 + · · · + rk Td = n are zero.
Remark. These formulae must be correctly interpreted when Ti contains no Dα and
(i)
(i)
(i)
rk Ti = n − 1. In that case, because of n − 1 = rk Ti = m1 + 2m2 + · · · + nmn , there
(i)
must be an ℓ, 1 ≤ ℓ ≤ n − 1, with mℓ ≥ 1. We then interpret the term
n − rk Ti − 1
1
(i)
(i)
n − rk Ti − 1 m(i)
1 , m2 , . . . , mn
as
n − rk Ti − 2
n − rk Ti − 1
1
1
= (i)
(i)
(i)
(i) ,
(i)
(i)
n − rk Ti − 1 m(i)
mℓ m1 , . . . , mℓ − 1, . . . , mn
1 , m2 , . . . , mn
where the multinomial coefficient is zero whenever
(i)
(i)
−1 = n − rk Ti − 2 < m1 + · · · + (mℓ − 1) + · · · + m(i)
n ,
(i)
(i)
(i)
except when all of m1 , . . . , mℓ − 1, . . . , mn are zero. Explicitly, one must read
n − rk Ti − 1
1
=0
(i)
(i)
n − rk Ti − 1 m(i)
1 , m2 , . . . , mn
20
C. KRATTENTHALER AND T. W. MÜLLER
if rk Ti = n − 1 but Ti 6= An−1 , and
n − rk Ti − 1
1
=1
(i)
(i)
n − rk Ti − 1 m(i)
1 , m2 , . . . , mn
if Ti = An−1 .
Proof of Theorem 9. Determining the decomposition number
NDn (T1 , T2 , . . . , Td ) = NDn (Td , . . . , T2 , T1 )
(recall (2.2)), respectively
NDcomb
(T1 , T2 , . . . , Td ) = NDcomb
(Td , . . . , T2 , T1 ),
n
n
amounts to counting all possible factorisations
(1, 2, . . . , n − 1, 1̄, 2̄, . . . , n − 1) (n, n̄) = σd · · · σ2 σ1 ,
(6.4)
where σi has type Ti as a parabolic Coxeter element, respectively has combinatorial
type Ti . Here also, the factorisation (6.4) is minimal in the sense that
n = ℓT (1, 2, . . . , n − 1, 1̄, 2̄, . . . , n − 1) (n, n̄) = ℓT (σ1 ) + ℓT (σ2 ) + · · · + ℓT (σd ),
since ℓT (σi ) = rk Ti , and since, by our assumption, the sum of the ranks of the Ti ’s
equals n.
We first prove Claim (iii). Let us assume, for contradiction, that there is a minimal
factorisation (6.4), in which, altogether, we find at least two type B cycles of length
≥ 2 in the (type B) disjoint cycle decompositions of the σi ’s. It can then be shown by
arguments similar to those in the proof of Claim (iii) in Theorem 7 that this leads to
a contradiction. Hence, “at worst,” we may find a type B cycle of length 1, (a, ā) say,
and another type B cycle, κ say. Both of them must be contained in the disjoint cycle
decomposition of one of the σi ’s since all the σi ’s are elements of W (Dn ). Given that
κ has length α − 1, the product of both, (a, ā) κ, is of combinatorial type Dα , α ≥ 2,
whereas, as a parabolic Coxeter element, it is of type Dα only if α ≥ 4. If α = 3, then
it is a parabolic Coxeter element of type A3 , and if α = 2 it is of type A21 . Thus, we are
actually in the cases to which Claims (i) and (ii) apply.
To prove Claim (i), we continue this line of argument. By a variation of the conjugation argument (5.4)–(5.6), we may assume that these two type B cycles are contained
in σd , σd = (a, ā) κ σd′ say, where, as above, (a, ā) is the type B cycle of length 1 and
κ is the other type B cycle, and where σd′ is free of type B cycles. In that case, (6.4)
takes the form
c = (1, 2, . . . , n − 1, 1̄, 2̄, . . . , n − 1) (n, n̄) = (a, ā) κ σd′ · · · σ1 .
(6.5)
If a 6= n, κ 6= (n, n̄), and if κ does not fix n, then (a, ā)κ 6≤T c, a contradiction. Likewise,
if a 6= n, κ = [b1 , b2 , . . . , bk ] with n ∈
/ {b1 , b2 , . . . , bk }, then (a, ā) κ 6≤T [1, 2, . . . , n − 1],
again a contradiction. Hence, we may assume that a = n, whence (a, ā) κ = κ (n, n̄)
forms a parabolic Coxeter element of type Dα , given that κ has length α − 1. We are
then in the position to determine all possible factorisations of the form (6.5), which
reduces to
(1, 2, . . . , n − 1, 1̄, 2̄, . . . , n − 1) = [1, 2, . . . , n − 1] = κσd′ · · · σ1 .
(6.6)
DECOMPOSITION NUMBERS FOR FINITE COXETER GROUPS
21
This is now a minimal type B factorisation of the form (5.3) with n replaced by n − 1.
We may therefore use Formula (5.1) with n replaced by n − 1, and with rk Tj replaced
by rk Tj − 1. These substitutions lead exactly to (6.1).
Finally, we turn to Claim (ii). First we discuss two degenerate cases which come
from the identifications D3 ∼ A3 , respectively D2 ∼ A21 , and which only occur for
(T1 , T2 ,
NDn (T1 , T2 , . . . , Td ) (but not for the combinatorial decomposition numbers NDcomb
n
. . . , Td )). It may happen that one of the factors in (6.4), let us say, without loss of
generality, σd , contains a type B cycle of length 1 and one of length 2 in its disjoint
cycle decomposition; that is, σd may contain
(n, n̄) [a, b] = (n, n̄) (a, b, ā, b̄) = [a, b] [b, n] [b, n̄].
As a parabolic Coxeter element, this is of type A3 . By the reduction (6.5)–(6.6), we
may count the number of these possibilities by Formula (5.1) with n replaced by n − 1,
(j)
(j)
rk Tj replaced by rk Tj −1, and m3 replaced by m3 −1. This explains the second term
in the factor in big parentheses on the right-hand side of (6.3). On the other hand, it
may happen that one of the factors in (6.4), let us say again, without loss of generality,
σd , contains two type B cycles of length 1 in its disjoint cycle decomposition; that is,
σd may contain (n, n̄) (a, ā). As a parabolic Coxeter element, this is of type A21 . By the
reduction (6.5)–(6.6), we may count the number of these possibilities by Formula (5.1)
(j)
(j)
with n replaced by n − 1, rk Tj replaced by rk Tj − 1, and m1 replaced by m1 − 2.
This explains the third term in the factor in big parentheses on the right-hand side of
(6.3).
From now on we may assume that none of the σi ’s contains a type B cycle in its (type
B) disjoint cycle decomposition. To determine the number of minimal factorisations
(6.4) in this case, we construct again a bijection between these factorisations and certain
maps. In what follows, we will still use the concept of a rotator, introduced in the
proof of Theorem 7. We apply again the procedure described in that proof. That
is, for each (ordinary) cycle (a1 , a2 , . . . , ak ) of σi , we create a k-gon coloured i, label
its vertices a1 , a2 , . . . , ak in clockwise order, and glue these polygons into a map by
identifying equally labelled vertices such that the rotator of each vertex is (1, 2, . . . , d).
However, this map can be embedded in the plane only if we allow the creation of an
inner face corresponding to the cycle (n, n̄) on the left-hand side of (6.4) (the outer face
corresponding to the large cycle (1, 2, . . . , n − 1, 1̄, 2̄, . . . , n − 1)). Moreover, this inner
face must be bounded by 2d edges. We call such a map, in which all faces except the
outer face and an inner face intersect only in vertices, and are coloured with colours
from {1, 2, . . . , d}, and in which the inner face is bounded by 2d edges, a d-atoll. For
example, if we take n = 10 and d = 3, and consider the factorisation
(1, 2, . . . , 9, 1̄, 2̄, . . . , 9̄) (10, 10) = σ3 σ2 σ1 ,
(6.7)
where σ3 = ((1, 4, 10, 7̄)), σ2 = ((1, 3)) ((4, 6, 10)) ((7, 8, 9)), and σ1 = ((1, 2)) ((4, 5)),
and apply this procedure, we obtain the 3-atoll in Figure 3. In the figure, the faces
corresponding to cycles are shaded. As in Figures 1 and 2, the outer face is not shaded.
Here, there is in addition an inner face which is not shaded, the face formed by the
vertices 4, 10, 4̄, 10. Again, the colours of the shaded faces are indicated by the numbers
1, 2, respectively 3, placed in the centre of the faces.
22
C. KRATTENTHALER AND T. W. MÜLLER
3
1
2̄
1
2
2 1
1̄
9
3
4̄
3
7
1
3
1
2 3
1
1
3
1 2
3
5̄
3̄
3
2
8̄
1
6̄
2
10
3
1
1
4
21
10
1
8
3
5
3
2
9̄
1
7̄
2
13
1
2
6
3
2
3
1
3
Figure 3. The 3-atoll corresponding to the factorisation (6.7)
Unsurprisingly, the fact that the result of the procedure can be realised as a d-atoll
follows again from Euler’s formula. More precisely, the number of faces corresponding
P P
P P
(i)
(i)
to the polygons is 2 di=1 nk=0 mk , the number of edges is 2 di=1 nk=0 mk (k + 1),
and the number of vertices is 2n. Hence, if we include the outer face and the inner face,
the number of vertices minus the number of edges plus the number of faces is
2n − 2
d X
n
X
i=1 k=0
(i)
mk (k
+ 1) + 2
d X
n
X
i=1 k=0
(i)
mk
+ 2 = 2n + 2 − 2
d X
n
X
(i)
k · mk
i=1 k=0
= 2n + 2 − 2 rk T1 − 2 rk T2 − · · · − 2 rk Td
= 2,
(6.8)
according to our assumption concerning the sum of the ranks of the types Ti .
Again, we may further simplify this geometric representation of a minimal factorisation (6.4) by deleting all vertex labels, marking the vertex which had label 1 with •,
and marking the vertex that had label n with . If this simplification is applied to the
3-atoll in Figure 3, we obtain the 3-atoll in Figure 4. Clearly, if drawn appropriately
into the plane, a d-atoll resulting from an application of the above procedure to a minimal factorisation (6.4) is symmetric with respect to a rotation by 180◦, the centre of the
rotation being the centre of the inner face; cf. Figure 4. As earlier, we shall abbreviate
DECOMPOSITION NUMBERS FOR FINITE COXETER GROUPS
3
3
1
2
1
3
1
2
3
2 1
3
1
2 3
1
1
3
1 2
3
1
2
1
1
21
3
3
2
1
3
1
2
13
2
1
3
23
3
Figure 4. A rotation-symmetric 3-atoll with two marked vertices
this property as rotation-symmetric. In fact, there is not much freedom for the choice of
the vertex marked by once a vertex has been marked by •. Clearly, if we run through
the vertex labelling process described in the proof of Theorem 7, labelling 1 the vertex
which is marked by •, we shall reconstruct the labels 1, 2, . . . , n − 1, 1̄, 2̄, . . . , n − 1.
This leaves only 2 vertices incident to the inner face unlabelled, one of which will have
to carry the mark .
In summary, under the assumptions of Claim (ii), the number of minimal factorisations (6.4), in which none of the σi ’s contains a type B cycle in its disjoint cycle
decomposition, equals twice the number of all rotation-symmetric d-atolls on 2n vertices, in which one vertex is marked by •, all vertices have rotator (1, 2, . . . , d)O , and
(i)
with exactly mk pairs of faces of colour i having k + 1 vertices, arranged symmetrically around the inner face (which is not coloured). Let us denote the number of these
d-atolls by ND′ n (T1 , T2 , . . . , Td ).
We must now enumerate these d-atolls. First of all, introducing a figure of speech,
we shall refer to coloured faces of a d-atoll which share an edge with the inner face but
not with the outer face as faces “inside the d-atoll,” and all others as faces “outside the
d-atoll.” For example, in Figure 4 we find two faces inside the 3-atoll, namely the two
loop faces attached to the vertices labelled 10, respectively 10, in Figure 3. Since, in a
d-atoll, the inner face is bounded by exactly 2d edges, inside the d-atoll, we find only
24
C. KRATTENTHALER AND T. W. MÜLLER
coloured faces containing exactly one vertex. Next, we travel counter-clockwise around
the inner face and record the coloured faces sharing an edge with both the inner and
outer faces. Thus we obtain a list of the form
F1 , F2 , . . . , Fℓ , Fℓ+1 , . . . , F2ℓ ,
where, except possibly for the marking, Fh+ℓ is an identical copy of Fh , h = 1, 2, . . . , ℓ.
In Figure 4, this list contains four faces, F̃1 , F̃2 , F̃3 , F̃4 , where F̃1 and F̃3 are the two
quadrangles of colour 3, and where F̃2 and F̃4 are the two triangles of colour 2 connecting
the two quadrangles.
Continuing the general argument, let the colour of Fh be ih . Inside the d-atoll, because
of the rotator condition, there must be {ih+1 −ih −1}d faces (containing just one vertex)
incident to the common vertex of Fh and Fh+1 coloured {ih +1}d , . . . , {ih+1 −1}d , where,
by definition,


if 0 ≤ x ≤ d
x,
{x}d := x + d, if x < 0

x − d, if x > d,
and where ih+ℓ = ih , h = 1, 2, . . . , ℓ. Here, if {ih + 1}d > {ih+1 − 1}d , the sequence
of colours {ih + 1}d , . . . , {ih+1 − 1}d must be interpreted “cyclically,” that is, as {ih +
1}d , {ih +1}d +1, . . . , d, 1, 2, . . . , {ih+1 −1}d . As we observed above, the number of edges
bounding the inner face is 2d. On the other hand, using the notation just introduced,
this number also equals
2
ℓ
X
h=1
ℓ
ℓ X
X
(ih+1 − ih ) + d · χ(ih+1 < ih ) = 2d
χ(ih+1 < ih ).
{ih+1 − ih }d = 2
h=1
h=1
Hence, there is precisely one h for which ih+1 < ih . Without loss of generality, we may
assume that h = ℓ, so that i1 < i2 < · · · < iℓ .
The ascending colouring of the faces F1 , F2 , . . . , Fℓ breaks the (rotation) symmetry
of the d-atoll. Therefore, we may first enumerate d-atolls without any marking, and
multiply the result by the number of all possible markings, which is n − 1. More
precisely, let ND′′ n (T1 , T2 , . . . , Td ) denote the number of all rotation-symmetric d-atolls
(i)
on 2n vertices, in which all vertices have rotator (1, 2, . . . , d)O , and with exactly mk
pairs of faces of colour i having k + 1 vertices, arranged symmetrically around the inner
face (which is not coloured). Then,
NDn (T1 , T2 , . . . , Td ) = 2ND′ n (T1 , T2 , . . . , Td )
= 2(n − 1)ND′′ n (T1 , T2 , . . . , Td ).
(6.9)
We use a generating function approach to determine ND′′ n (T1 , T2 , . . . , Td ), which requires a combinatorial decomposition of our objects. Let G(z) be the generating function
X
G(z) =
w(A),
(6.10)
A∈A
DECOMPOSITION NUMBERS FOR FINITE COXETER GROUPS
25
where A is the set of all rotation-symmetric d-atolls, in which all vertices have rotator
(1, 2, . . . , d)O , and where
w(A) =
d
Y
1
#(faces
2
zi
of A with colour i)
i=1
d Y
∞
Y
1
#(faces of A with colour i and k vertices)
2
pi,k
.
i=1 k=1
Here, z = (z1 , z2 , . . . , zd ), with the zi ’s, i = 1, 2, . . . , d, and the pi,k , i = 1, 2, . . . , d,
k = 1, 2, . . . , being indeterminates. Clearly, in view of the bijection between minimal
factorisations (6.4) and d-atolls described earlier, and by (6.9), we have
+
*
d Y
n
Y
(i)
m
k
G(z),
(6.11)
pi,k+1
NDn (T1 , T2 , . . . , Td ) = 2(n − 1) zc
i=1 k=0
where c = (c1 , c2 , . . . , cd ), with ci equal to the number of type A cycles of σi ; that is,
P
(i)
ci = nk=0 mk , i = 1, 2, . . . , d. Here, and in the sequel, we use the multi-index notation
introduced at the beginning of Section 3. For later use, we observe that, for all i, ci is
related to rk Ti via
ci = n − rk Ti .
(6.12)
Now, let A be a d-atoll in A such that the faces which share an edge with both the
inner and outer faces are
F1 , F2 , . . . , Fℓ , Fℓ+1 , . . . , F2ℓ ,
where Fh+ℓ is an identical copy of Fh , where the colour of Fh is ih , h = 1, 2, . . . , ℓ, and
with i1 < i2 < · · · < iℓ . We decompose A by separating from each other the polygons
which touch in vertices of the inner face. The decomposition in the case of our example
in Figure 4 is shown in Figure 5. Ignoring identical copies which are there due to the
rotation symmetry, we obtain a list
(1)
(1)
(1)
(1)
(1)
(1)
K1 , Li1 +1 , . . . , Li2 −1 , Ci2 +1 , . . . , Cd , C1 , . . . , Ci1 −1 ,
(2)
(2)
(2)
(2)
(2)
(2)
K2 , Li2 +1 , . . . , Li3 −1 , Ci3 +1 , . . . , Cd , C1 , . . . , Ci2 −1 , . . .
(ℓ)
(ℓ)
(ℓ)
(ℓ)
(ℓ)
(ℓ)
Kℓ , Liℓ +1 , . . . , Ld , L1 , . . . , Li1 −1 , Ci1 +1 , . . . , Ciℓ −1 , (6.13)
where Kh is the d-cactus containing the face Fh , and, hence, a d-cactus in which all
but two neighbouring vertices have rotator (1, 2, . . . , d)O , the latter two vertices being
(h)
incident to just one face, which is of colour ih , where Lj is a face of colour j with
(h)
just one vertex, and where Cj is a d-cactus in which all but one vertex have rotator
(1, 2, . . . , d)O , the distinguished vertex being incident to just one face, which is of colour
j, h = 1, 2, . . . , ℓ and j = 1, 2, . . . , d. With this notation, our example in Figure 5 is
one in which ℓ = 2, i1 = 2, i2 = 3.
The d-cacti Kh can be further decomposed. Namely, assuming that the face Fh
is a k-gon (of colour ih ), let C1 , C2 , . . . , Ck−2 be the d-cacti incident to this k-gon,
read in clockwise order, starting with the d-cactus to the left of the two distinguished
vertices. Figure 6 illustrates this further decomposition of the d-cactus K2 from Figure 5. After removal of Fh , we are left with the ordered collection C1 , C2 , . . . , Ck−2
of d-cacti, each of which having the property that the rotator of all but one vertex is
(1, 2, . . . , d)O , the exceptional vertex having rotator (1, . . . , ih − 1, ih + 1, . . . , d)O . By
separating from each other the polygons of colours 1, . . . , ih − 1, ih + 1, . . . , d which
26
C. KRATTENTHALER AND T. W. MÜLLER
1
3
1 2
(2)
L1
1
3
2
13
2
2
(1)
1
1
3
3
1
1
K2
K1
C1
3
3
2
Figure 5. The decomposition of the 3-atoll in Figure 4
touch in the exceptional vertex, each d-cactus Ci in turn can be decomposed into dcacti Ci,1 , . . . , Ci,ih−1 , Ci,ih+1 , . . . , Ci,d with Ci,j ∈ Cj for all k, where Cj denotes the set
of all d-cacti in which all but one vertex have rotator (1, 2, . . . , d)O , the distinguished
vertex being incident to just one face, which is of colour j.
Let ωj (z) denote the generating function for the d-cacti in Cj , that is,
X
ωj (z) =
w(C).
(6.14)
C∈Cj
Furthermore, for i = 1, 2, . . . , d, define the formal power series Pi (u) in one variable u
via
∞
X
Pi (u) =
pi,k uk−1.
k=1
Then, by the decomposition (6.13) and the further decomposition of the Kh ’s that we
just described, the contribution of the above d-atolls to the generating function (6.10)
DECOMPOSITION NUMBERS FOR FINITE COXETER GROUPS
1
3
1 2
3
27
3
C1
1
2
13
2
1
C2
3
Figure 6. The decomposition of K2 in Figure 5
is
ℓ
Y
!
ω1 (z) · · · ωd (z)
Pij
− pij ,1
ωij (z)
!
Qd
(ω1 (z) · · · ωd (z))ℓ−1
j=1 zj pj,1
× Qℓ
Qℓ
j=1 zij pij ,1
j=1 ωij (z)


! ℓ
ω1 (z)···ωd (z)
d
Y
ωij (z)
zj pj,1 Y  Pij
− 1 ,
ω
(z)
p
j
i
,1
j
j=1
j=1
zij ωij (z)
ω (z) · · · ωd (z)
j=1 1
=
the term in the first line corresponding to the contribution of the Kj ’s, the first term in
(j)
the second line corresponding to the contribution of the Lk ’s, and the second term in
(j)
the second line corresponding to the contribution of the Ck ’s. These expressions must
be summed over ℓ = 2, 3, . . . , d and all possible choices of 1 ≤ i1 < i2 < · · · < iℓ ≤ d to
28
C. KRATTENTHALER AND T. W. MÜLLER
obtain the desired generating function G(z), that is,


! d
ω1 (z)···ωd (z)
ℓ
d
P
X
Y
Y zj pj,1 X
ij
ωij (z)

− 1
G(z) =
ω
(z)
p
,
1
j
i
j
j=1
ℓ=2 1≤i1 <i2 <···<iℓ ≤d j=1

 ω (z)···ω (z) 

! d
ω1 (z)···ωd (z)
1
d
d
d
P
Y Pj
X
Y zj pj,1
j
ωj (z)
ωj (z)


=
−
− 1 − 1 .
ω
(z)
p
p
j
j,1
j,1
j=1
j=1
j=1
(6.15)
Here we have used the elementary identity
d
X
X
Xi1 Xi2 · · · Xiℓ = (1 + X1 )(1 + X2 ) · · · (1 + Xd ).
ℓ=0 1≤i1 <i2 <···<iℓ ≤d
Before we are able to proceed, we must find functional equations for the generating
functions ωj (z), j = 1, 2, . . . , d. Given a d-cactus C in Cj such that the distinguished
vertex is incident to a k-gon (of colour j), we decompose it in a manner analogous
to the decomposition of Kh above. To be more precise, let C1 , C2 , . . . , Ck−1 be the
d-cacti incident to this k-gon, read in clockwise order, starting with the d-cactus to
the left of the distinguished vertex. After removal of the k-gon, we are left with the
ordered collection C1 , C2 , . . . , Ck−1 of d-cacti, each of which having the property that
the rotator of all but one vertex is (1, 2, . . . , d)O , the exceptional vertex having rotator
(1, . . . , j − 1, j + 1, . . . , d)O . By separating from each other the polygons of colours
1, . . . , j − 1, j + 1, . . . , d which touch in the exceptional vertex, each d-cactus Ci in turn
can be decomposed into d-cacti Ci,1, . . . , Ci,j−1, Ci,j+1, . . . , Ci,d with Ci,k ∈ Ck for all k.
The upshot of these combinatorial considerations is that
ωj (z) = zj Pj (ω1 (z) · · · ωd (z)/ωj (z)),
j = 1, 2, . . . , d,
or, equivalently,
zj =
ωj (z)
,
Pj (ω1 (z) · · · ωd (z)/ωj (z))
j = 1, 2, . . . , d.
Using this relation, the expression (6.15) for G(z) may now be further simplified, and
we obtain
ω1 (z)···ωd (z)
d P
d
d
X
Y
Y
j
ωj (z)
p
pj,1
.
j,1
+ (d − 1)
G(z) = 1 −
ω1 (z)···ωd (z)
ω1 (z)···ωd (z)
p
j,1
P
P
j=1
j=1
j=1
j
j
ωj (z)
ωj (z)
This is substituted in (6.11), to obtain
NDn (T1 , T2 , . . . , Td )
*
d Y
n
Y
+ d
Y
(i)
mk

pi,k+1
pj,1

d P
X
j
ω1 (z)···ωd (z)
ωj (z)

ω1 (z)···ωd (z)
pj,1
P
j=1
j=1 j
i=1 k=0
ωj (z)
+ d
*
d Y
n
Y
Y
(i)
p
m
k
j,1
. (6.16)
pi,k+1
+ 2(n − 1)(d − 1) zc
ω1 (z)···ωd (z)
j=1 Pj
i=1 k=0
ωj (z)
= −2(n − 1) zc
DECOMPOSITION NUMBERS FOR FINITE COXETER GROUPS
29
Now the problem is set up for application of the Lagrange–Good inversion formula.
Let fi (z) = zi /Pi(z1 · · · zd /zi ), i = 1, 2, . . . , d. If we substitute fi (z) in place of zi ,
i = 1, 2, . . . , d, in (6.16), and apply Theorem 1 with

d
Y

g(z) =
j=1
Pj
p
j,1
z1 ···zd
zj


d
X
Pj
z1 ···zd
zj
pj,1
j=1
,
respectively
d
Y
g(z) =
j=1
Pj
p
j,1
z1 ···zd
zj
,
we obtain that
NDn (T1 , T2 , . . . , Td )
+ d
! d
*
d Y
n
Y
X
Y
(i)
m
k
pi,1
pi,k+1
= −2(n − 1) z0
i=1
i=1 k=0
*
+ 2(n − 1)(d − 1) z
0
d Y
n
Y
i=1 k=0
(i)
mk
pi,k+1
+
j=1
d
Y
i=1
pi,1
!
∂fi
f (z) det
(z)
1≤i,k≤d ∂zk
∂fi
−c
f (z) det
(z) , (6.17)
1≤i,k≤d ∂zk
zj
fj (z)pj,1
!
−c
where 0 stands for the vector (0, 0, . . . , 0). We treat the two terms on the right-hand
side of (6.17) separately. We begin with the second term:
*
+
!
∂fi
pi,1 f (z) det
z
(z)
1≤i,k≤d ∂zk
i=1
i=1 k=0


z
·
·
·
z


1
d
c
−1
i

,
i = k
Pi








zi
+ d
!
*



d
n


Y
Y Y m(i)


z
·
·
·
z
1
d
c
−2
c
i
k

−P
pi,1
det 
pi,k+1
= z
i

1≤i,k≤d 


z
i



i=1
i=1 k=0





z1 · · · zd z1 · · · zd


′


, i=
6 k
 ×Pi
zi
zk
+
!
*
d
d Y
n
Y
Y
(i)
mk
pi,1
pi,k+1
= zc
0
d Y
n
Y
(i)
mk
pi,k+1
i=1 k=0
d
Y
−c
i=1

z1 · · · zd
c
−1

i
 Pi
,


zi

× det 
1
d ci −1
1≤i,k≤d 

(u) 
− ci − 1 u du Pi
u=z1 ···zd /zi


i = k



.



, i 6= k 

30
C. KRATTENTHALER AND T. W. MÜLLER
Reading coefficients, we obtain
d Y
i=1
ci − 1
(i)
(i)
(i)
m1 , m2 , . . . , mn


 1,
i = k

rk Ti
det 
, i 6= k 
1≤i,k≤d
−
ci − 1
d Y
ci − 1
n−1
1 − χ(i 6= k)
=
det
,
(i)
(i)
(i)
ci − 1
m1 , m2 , . . . , mn 1≤i,k≤d
i=1
the second line being due to (6.12). Now we can apply Lemma 2 with Xi = ci − 1 and
Yi = n − 1, i = 1, 2, . . . , d. The term
d
X
i=1
Xi −
d
X
Yi =
i=2
d
X
(ci − 1) − (d − 1)(n − 1)
i=1
=
d
X
(n − rk Ti − 1) − (d − 1)(n − 1)
i=1
on the right-hand side of (3.2) simplifies to −1 due to our assumption concerning the
sum of the ranks of the types Ti . Hence, if we use the relation (6.12) once more, the
second term on the right-hand side of (6.17) is seen to equal
−2(d − 1)(n − 1)
d
d
Y
i=1
n − rk Ti − 1
1
(i)
(i) .
n − rk Ti − 1 m(i)
1 , m2 , . . . , mn
This explains the second term in the factor in big parentheses in (6.2) and the fourth
term in the factor in big parentheses on the right-hand side of (6.3).
Finally, we come to the first term on the right-hand side of (6.17). We have
+ d
!
*
d Y
n
Y
Y
(i)
zj
∂fi
m
0
−c
k
pi,1
pi,k+1
z
f (z) det
(z)
1≤i,k≤d ∂zk
f
(z)p
j
j,1
i=1
i=1 k=0


z
·
·
·
z


1
d
c
−1+χ(i=j)
i



,
i = k

Pi




z
+
*
i


d
d
n


Y Y m(i) Y 


z
·
·
·
z
1
d
c
−2+χ(i=j)
c
i
k

 pi,1  det  −P
= z
pi,k+1
i

 1≤i,k≤d 


z
i




i=1
i=1 k=0






z
·
·
·
z
z
·
·
·
z


1
d
1
d
i6=j

, i 6= k 

 ×Pi′
zi
zk


+
*
d
d Y
n
Y
(i)

Y
mk
c

pi,k+1  pi,1 
= z

i=1 k=0
i=1
i6=j

z1 · · · zd
ci −1+χ(i=j)

Pi
,


zi

× det 
1
d ci −1+χ(i=j)
1≤i,k≤d 

(u) 
− ci − 1 + χ(i = j) u du Pi
u=z1 ···zd /zi


i = k



.



, i 6= k 

DECOMPOSITION NUMBERS FOR FINITE COXETER GROUPS
31
Reading coefficients, we obtain
i=1


det 
1≤i,k≤d
−

1,
i = k

rk Ti
(i)
(i)
(i)
, i 6= k 
m1 , m2 , . . . , mn
ci − 1 + χ(i = j)
d
Y ci − 1 + χ(i = j) 1 − χ(j 6= k) cnj , i = j
,
=
det
(i)
(i)
(i)
1 − χ(i 6= k) cn−1
, i=
6 j
1≤i,k≤d
m
,
m
,
.
.
.
,
m
n
−1
1
2
i
i=1
d Y
ci − 1 + χ(i = j)
the second line being due to (6.12). Now we can apply Corollary 3 with r = j, Xi =
ci − 1, i = 1, . . . , j − 1, j + 1, . . . , d, Xj = cj , Y = n − 1, and Z = n. The term
!
d
d
X
X
(ci − 1) − cj − (d − 1)(n − 1)n
Xi + (Y − Z)Xj − (d − 1)Y Z = n 1 +
Z
i=1
i=1
d
X
=n
(n − rk Ti − 1) + n − cj − (d − 1)(n − 1)n
i=1
on the right-hand side of (3.2) simplifies to −cj due to our assumption concerning the
sum of the ranks of the types Ti . Hence, if we use the relation (6.12) once more, the
second term on the right-hand side of (6.17) is seen to equal the sum over j = 1, 2, . . . , d
of
Y
d
n − rk Tj
n − rk Ti − 1
1
d−1
2(n − 1)
(i)
(i)
(i) .
(j)
(j)
(j)
m1 , m2 , . . . , mn i=1 n − rk Ti − 1 m1 , m2 , . . . , mn
i6=j
This explains the first terms in the factors in big parentheses on the right-hand sides
of (6.2) and (6.3).
The proof of the theorem is complete.
Combining the previous theorem with the summation formula of Lemma 4, we can
now derive compact formulae for all type Dn decomposition numbers.
Theorem 10. (i) Let the types T1 , T2 , . . . , Td be given, where
(i)
m
(i)
(i)
m
n
Ti = A1 1 ∗ A2 2 ∗ · · · ∗ Am
n ,
i = 1, 2, . . . , j − 1, j + 1, . . . , d,
and
(j)
m
(j)
m
(j)
n
Tj = Dα ∗ A1 1 ∗ A2 2 ∗ · · · ∗ Am
,
n
for some α ≥ 2. Then
n−1
= (n − 1)
rk T1 + rk T2 + · · · + rk Td − 1
Y
d
n − rk Tj
n − rk Ti − 1
1
×
(i)
(i)
(i) , (6.18)
(j)
(j)
(j)
m1 , m2 , . . . , mn i=1 n − rk Ti − 1 m1 , m2 , . . . , mn
(T1 , T2 , . . . , Td )
NDcomb
n
d−1
i6=j
where the multinomial coefficient is defined as in Lemma 4. For α ≥ 4, the number
NDn (T1 , T2 , . . . , Td ) is given by the same formula.
32
C. KRATTENTHALER AND T. W. MÜLLER
(ii) Let the types T1 , T2 , . . . , Td be given, where
(i)
(i)
m
(i)
m
n
Ti = A1 1 ∗ A2 2 ∗ · · · ∗ Am
n ,
i = 1, 2, . . . , d.
Then
(T1 , T2 , . . . , Td )
NDcomb
n
d−1
= (n − 1)

n−1
rk T1 + rk T2 + · · · + rk Td − 1
Y
!
d
d  X
n − rk Tj
n
−
rk
T
−
1
1
i
×
(i)
(i)
(j)
(j)
(j)
2
n − rk Ti − 1 m(i)
m1 , m2 , . . . , mn
1 , m2 , . . . , mn
i=1
j=1
i6=j


P
P
n − dℓ=1 rk Tℓ n − 1 − dℓ=1 rk Tℓ
+
− 2(d − 2)(n − 1)
Pd
rk
T
ℓ
ℓ=1

d
Y

n − rk Ti − 1
1

·
(i)
(i)
(i)  , (6.19)
n
−
rk
T
−
1
m
,
m
,
.
.
.
,
m
n
i
1
2
i=1
whereas
d−1
NDn (T1 , T2 , . . . , Td ) = (n − 1)

n−1
rk T1 + rk T2 + · · · + rk Td − 1
d
d
Y
X
n
−
rk
T
−
1
1
i
×
(i)
(i)

n − rk Ti − 1 m(i)
1 , m2 , . . . , mn
j=1
i=1
i6=j
+
n − rk Tj
!
2
n − rk Tj
(j)
(j)
(j)
m1 , m2 , . . . , mn
n − rk Tj
!
(j)
(j)
(j)
(j)
(j)
(j)
(j)
(j) +
m1 − 2, m2 , . . . , mn
m1 , m2 , m3 − 1, m4 , . . . , mn


P
P
n − dℓ=1 rk Tℓ n − 1 − dℓ=1 rk Tℓ
+
− 2(d − 2)(n − 1)
Pd
rk
T
ℓ
ℓ=1

d
Y

n − rk Ti − 1
1

·
(i)
(i)
(i)  . (6.20)
n
−
rk
T
−
1
m
,
m
,
.
.
.
,
m
n
i
1
2
i=1
(T1 , T2 , . . . , Td )
(iii) All other decomposition numbers NDn (T1 , T2 , . . . , Td ) and NDcomb
n
are zero.
Remark. The caveats on interpretations of the formulae in Theorem 9 for critical choices
of the parameters (cf. the Remark after the statement of that theorem) apply also to
the formulae of Theorem 10.
DECOMPOSITION NUMBERS FOR FINITE COXETER GROUPS
33
Proof. We proceed in a manner similar to the proof of Theorem 8. If we write r for
n − rk T1 − rk T2 − · · · − rk Td and set Φ = Dn , then the relation (2.3) becomes
X
(6.21)
NDn (T1 , T2 , . . . , Td , T ),
NDn (T1 , T2 , . . . , Td ) =
T :rk T =r
in place of NDn .
with the same relation holding for NDcomb
n
m2
mn
1
In order to prove (6.18), we let T = Am
1 ∗ A2 ∗ · · · ∗ An and use (6.1) in (6.21), to
obtain
X
1
n−r−1
comb
d
NDn (T1 , T2 , . . . , Td ) =
(n − 1)
n − r − 1 m1 , m2 , . . . , mn
m1 +2m2 +···+nmn =r
Y
d
n − rk Tj
n − rk Ti − 1
1
·
(i)
(i)
(i) .
(j)
(j)
(j)
m1 , m2 , . . . , mn i=1 n − rk Ti − 1 m1 , m2 , . . . , mn
i6=j
If we use (3.4) with M = n − r − 1, we arrive at our claim after little simplification.
Next we prove (6.19). In contrast to the previous argument, here the summation
on the right-hand side of (6.21) must be taken over all types T of the form T =
m2
m1
m2
mn
mn
1
Dα ∗ Am
1 ∗ A2 ∗ · · · ∗ An , α ≥ 2, as well as of the form T = A1 ∗ A2 ∗ · · · ∗ An .
For the sum over the former types, we have to substitute (6.1) in (6.21), to get
n
X
X
d
(n − 1)
α=2 m1 +2m2 +···+nmn =r−α
n−r
m1 , m2 , . . . , mn
n − rk Ti − 1
1
·
(i)
(i)
(i) . (6.22)
n
−
rk
T
−
1
m
,
m
,
.
.
.
,
m
n
i
1
2
i=1
d
Y
On the other hand, for the sum over the latter types, we have to substitute (6.2) in
(6.21), to get
Y
d
n − rk Ti − 1
1
n−r
2
(n − 1)
(i)
(i)
m1 , m2 , . . . , mn i=1 n − rk Ti − 1 m(i)
1 , m2 , . . . , mn
m1 +2m2 +···+nmn =r
X
1
n−r−1
d
+
(n − 1)
n − r − 1 m1 , m2 , . . . , mn
m1 +2m2 +···+nmn =r

Y
d
d  X
n − rk Tj
n − rk Ti − 1
1

· 2
(i)
(i)
(i)
(j)
(j)
(j)
m1 , m2 , . . . , mn i=1 n − rk Ti − 1 m1 , m2 , . . . , mn
j=1
X
d
i6=j

d
Y

n − rk Ti − 1
1

−2(d − 1)(n − 1)
(i)
(i)
(i)  . (6.23)
n
−
rk
T
−
1
m
,
m
,
.
.
.
,
m
n
i
1
2
i=1
34
C. KRATTENTHALER AND T. W. MÜLLER
We simplify (6.22) by using (3.4) with r replaced by r − α and M = n − r, and by
subsequently applying the elementary summation formula
X
n n X
n−2
n−2
n−α−1
n−α−1
.
(6.24)
=
=
=
r
−
2
n
−
r
n
−
r
−
1
r
−
α
α=2
α=2
The expression which we obtain in this way explains the fraction in the third line of
(6.19) multiplied by the expression in the last line. On the other hand, we simplify the
sums in (6.23) by using (3.4) with M = n − r, respectively M = n − r − 1. Thus, the
expression (6.23) becomes
d
n − rk Ti − 1
1
n−1 Y
2(n − 1)
(i)
(i)
r
n − rk Ti − 1 m(i)
1 , m2 , . . . , mn
i=1
Y
X
d
d n − rk Tj
n − rk Ti − 1
1
d−1 n − 1
2
+(n−1)
(i)
(i)
(i)
(j)
(j)
(j)
r
m1 , m2 , . . . , mn i=1 n − rk Ti − 1 m1 , m2 , . . . , mn
j=1
d
i6=j
!
n − rk Ti − 1
1
,
(i)
(i)
n − rk Ti − 1 m(i)
1 , m2 , . . . , mn
i=1
d
Y
− 2(d − 1)(n − 1)
which explains the expression in the second line of (6.19) and the second expression in
the third line of (6.19) multiplied by the expression in the last line.
The proof of (6.20) is analogous, using (6.3) instead of (6.2). We leave the details to
the reader.
7. Generalised non-crossing partitions
In this section we recall the definition of Armstrong’s [1] generalised non-crossing
partitions poset, and its combinatorial realisation from [1] and [29] for the types An ,
Bn , and Dn .
Let again Φ be a finite root system of rank n, and let W = W (Φ) be the corresponding
reflection group. We define first the non-crossing partition lattice NC(Φ) (cf. [8, 15]).
Let c be a Coxeter element in W . Then NC(Φ) is defined to be the restriction of the
partial order ≤T from Section 2 to the set of all elements which are less than or equal
to c in this partial order. This definition makes sense since any two Coxeter elements
in W are conjugate to each other; the induced inner automorphism then restricts to an
isomorphism of the posets corresponding to the two Coxeter elements. It can be shown
that NC(Φ) is in fact a lattice (see [16] for a uniform proof), and moreover self-dual (this
is obvious from the definition). Clearly, the minimal element in NC(Φ) is the identity
element in W , which we denote by ε, and the maximal element in NC(Φ) is the chosen
Coxeter element c. The term “non-crossing partition lattice” is used because NC(An )
is isomorphic to the lattice of non-crossing partitions of {1, 2, . . . , n + 1}, originally
introduced by Kreweras [30] (see also [20] and below), and since also NC(Bn ) and
NC(Dn ) can be realised as lattices of non-crossing partitions (see [4, 32] and below).
In addition to a fixed root system, the definition of Armstrong’s generalised noncrossing partitions requires a fixed positive integer m. The poset of m-divisible noncrossing partitions associated to the root system Φ has as ground-set the following subset
DECOMPOSITION NUMBERS FOR FINITE COXETER GROUPS
35
of (NC(Φ))m+1 :
NC m (Φ) = (w0 ; w1 , . . . , wm ) : w0 w1 · · · wm = c and
The order relation is defined by
ℓT (w0 ) + ℓT (w1 ) + · · · + ℓT (wm ) = ℓT (c) . (7.1)
(u0 ; u1 , . . . , um) ≤ (w0 ; w1 , . . . , wm ) if and only if ui ≥T wi , 1 ≤ i ≤ m.
(According to this definition, u0 and w0 need not be related in any way. However, it
follows from [1, Lemma 3.4.7] that, in fact, u0 ≤T w0 .) The poset NC m (Φ) is graded
by the rank function
rk (w0 ; w1 , . . . , wm ) = ℓT (w0 ).
(7.2)
Thus, there is a unique maximal element, namely (c; ε, . . . , ε), where ε stands for the
identity element in W , but, for m > 1, there are many different minimal elements. In
particular, NC m (Φ) has no least element if m > 1; hence, NC m (Φ) is not a lattice for
m > 1. (It is, however, a graded join-semilattice, see [1, Theorem 3.4.4].)
In what follows, we shall use the notions “generalised non-crossing partitions” and
“m-divisible non-crossing partitions” interchangeably, where the latter notion will be
employed particularly in contexts in which we want to underline the presence of the
parameter m.
In the remainder of this section, we explain combinatorial realisations of the mdivisible non-crossing partitions of types An−1 , Bn , and Dn . In order to be able to do
so, we need to recall the definition of Kreweras’ non-crossing partitions of {1, 2, . . . , N},
his “partitions non croisées d’un cycle” of [30]. We place N vertices around a cycle,
and label them 1, 2, . . . , N in clockwise order. The circular representation of a partition
of the set {1, 2, . . . , N} is the geometric object which arises by representing each block
{i1 , i2 , . . . , ik } of the partition, where i1 < i2 < · · · < ik , by the polygon consisting of
the vertices labelled i1 , i2 , . . . , ik and edges which connect these vertices in clockwise
order. A partition of {1, 2, . . . , N} is called non-crossing if any two edges in its circular
representation are disjoint. Figure 7 shows the non-crossing partition
{{1, 2, 21}, {3, 19, 20}, {4, 5, 6}, {7, 17, 18}, {8, 9, 10, 11, 12, 13, 14, 15, 16}}
of {1, 2, . . . , 21}. There is a natural partial order on Kreweras’ non-crossing partitions
defined by refinement: a partition π1 is less than or equal to the partition π2 if every
block of π1 is contained in some block of π2 .
If Φ = An−1 , the m-divisible non-crossing partitions are in bijection with Krewerastype non-crossing partitions of the set {1, 2, . . . , mn}, in which all the block sizes are
g m (An−1 ). It
divisible by m. We denote the latter set of non-crossing partitions by NC
has been first considered by Edelman in [18]. In fact, Figure 7 shows an example of a
3-divisible non-crossing partition of type A20 .
Given an element (w0 ; w1 , . . . , wm ) ∈ NC m (An−1 ), the bijection, ▽m
An−1 say, from [1,
Theorem 4.3.8] works by “blowing up” w1 , w2 , . . . , wm , thereby “interleaving” them,
and then “gluing” them together by an operation which is called Kreweras complement
in [1]. More precisely, for i = 1, 2, . . . , m, let τm,i be the transformation which maps a
permutation w ∈ Sn to a permutation τm,i (w) ∈ Smn by letting
(τm,i (w))(mk + i − m) = mw(k) + i − m,
k = 1, 2, . . . , n,
36
C. KRATTENTHALER AND T. W. MÜLLER
21
1
20
2
3
19
4
18
5
17
6
16
7
15
8
14
9
13
12
11
10
Figure 7. Combinatorial realisation of a 3-divisible non-crossing partition of type A6
and (τm,i (w))(l) = l for all l 6≡ i (mod m). At this point, the reader should recall from
Section 2 that W (An−1 ) is the symmetric group Sn , and that the standard choice of
a Coxeter element in W (An−1 ) = Sn is c = (1, 2, . . . , n). With this choice of Coxeter
element, the announced bijection maps (w0 ; w1 , . . . , wm ) ∈ NC m (An−1 ) to
−1
▽m
(τm,2 (w2 ))−1 · · · (τm,m (wm ))−1 .
An−1 (w0 ; w1 , . . . , wm ) = (1, 2, . . . , mn) (τm,1 (w1 ))
We refer the reader to [1, Sec. 4.3.2] for the details. For example, let n = 7, m = 3,
w0 = (4, 5, 6), w1 = (3, 6), w2 = (1, 7), and w3 = (1, 2, 6). Then (w0 ; w1 , w2 , w3 ) is
mapped to
▽3A6 (w0 ; w1, w2 , w3 ) = (1, 2, . . . , 21) (7, 16) (2, 20) (18, 6, 3)
= (1, 2, 21) (3, 19, 20) (4, 5, 6) (7, 17, 18) (8, 9, . . . , 16).
(7.3)
Figure 7 shows the graphical representation of (7.3) on the circle, in which we represent
a cycle (i1 , i2 , . . . , ik ) as a polygon consisting of the vertices labelled i1 , i2 , . . . , ik and
edges which connect these vertices in clockwise order.
It is shown in [1, Theorem 4.3.8] that ▽m
An−1 is in fact an isomorphism between the
g m (An−1 ). Furthermore, it is proved in [1, Theorem 4.3.13]
posets NC m (An−1 ) and NC
DECOMPOSITION NUMBERS FOR FINITE COXETER GROUPS
14
15
1
37
2
3
13
4
12
5
11
6
10
7
9
8
8
9
7
6
10
11
5
12
4
13
3
2
1
15
14
Figure 8. Combinatorial realisation of a 3-divisible non-crossing partition of type B5
that
ci (w0 ) = bi (▽m
An−1 (w0 ; w1 , . . . , wm )),
i = 1, 2, . . . , n,
(7.4)
where ci (w0 ) denotes the number of cycles of length i of w0 and bi (π) denotes the
number of blocks of size mi in the non-crossing partition π.
If Φ = Bn , the m-divisible non-crossing partitions are in bijection with Kreweras-type
non-crossing partitions π of the set {1, 2, . . . , mn, 1̄, 2̄, . . . , mn}, in which all the block
sizes are divisible by m, and which have the property that if B is a block of π then also
B := {x̄ : x ∈ B} is a block of π. (Here, as earlier, we adopt the convention that x̄¯ = x
g m (Bn ). A block
for all x.) We denote the latter set of non-crossing partitions by NC
g m (Bn ) can only
B with B = B is called a zero block . A non-crossing partition in NC
have at most one zero block. Figures 8 and 9 give examples of 3-divisible non-crossing
partitions of type B5 . Figure 8 shows one without a zero block, while Figure 9 shows
one with a zero block. Clearly, the condition that B is a block of the partition if and
only if B is a block translates into the condition that the geometric realisation of the
partition is invariant under rotation by 180◦.
38
C. KRATTENTHALER AND T. W. MÜLLER
14
15
1
2
3
13
4
12
5
11
6
10
7
9
8
8
9
7
6
10
11
5
12
4
13
3
2
1
15
14
Figure 9. A 3-divisible non-crossing partition of type B5 with zero block
Given an element (w0 ; w1, . . . , wm ) ∈ NC m (Bn ), the bijection, ▽m
Bn say, from [1,
m
Theorem 4.5.6] works in the same way as for NC (An−1 ). That is, recalling from
Section 2 that W (Bn ) can be combinatorially realised as a subgroup of the group
of permutations of {1, 2, . . . , n, 1̄, 2̄, . . . , n̄}, and that, in this realisation, the standard
choice of a Coxeter element is c = [1, 2, . . . , n] = (1, 2, . . . , n, 1̄, 2̄, . . . , n̄), the announced
bijection maps (w0 ; w1 , . . . , wm ) ∈ NC m (Bn ) to
−1
(τ̄m,2 (w2 ))−1 · · · (τ̄m,m (wm ))−1 ,
▽m
Bn (w0 ; w1 , . . . , wm ) = [1, 2, . . . , mn] (τ̄m,1 (w1 ))
where τ̄m,i is the obvious extension of the above transformations τm,i : namely we let
(τ̄m,i (w))(mk + i − m) = mw(k) + i − m, k = 1, 2, . . . , n, 1̄, 2̄, . . . , n̄,
and (τ̄m,i (w))(l) = l and (τ̄m,i (w))(¯l) = l¯ for all l 6≡ i (mod m), where mk̄ + i − m is
identified with mk + i − m for all k and i. We refer the reader to [1, Sec. 4.5] for the
details. For example, let n = 5, m = 3, w0 = ((2, 4)), w1 = [1] = (1, 1̄), w2 = ((1, 4)),
and w3 = ((2, 3)) ((4, 5)). Then (w0 ; w1 , w2 , w3 ) is mapped to
▽3B5 (w0 ; w1 , w2 , w3) = [1, 2, . . . , 15] [1] ((2, 11)) ((6, 9)) ((12, 15))
= ((1, 2̄, 12)) ((3, 4, 5, 6, 10, 11)) ((7, 8, 9)) ((13, 14, 15)).
(7.5)
DECOMPOSITION NUMBERS FOR FINITE COXETER GROUPS
39
Figure 8 shows the graphical representation of (7.5).
It is shown in [1, Theorem 4.5.6] that ▽m
Bn is in fact an isomorphism between
m
m
g
the posets NC (Bn ) and NC (Bn ). Furthermore, it is proved in [1, proof of Theorem 4.3.13] that
ci (w0 ) = bi (▽m
Bn (w0 ; w1 , . . . , wm )),
i = 1, 2, . . . , n,
(7.6)
where ci (w0 ) denotes the number of type A cycles (recall the corresponding terminology
from Section 4) of length i of w0 and bi (π) denotes one half of the number of non-zero
blocks of size mi in the non-crossing partition π. (Recall that non-zero blocks come in
“symmetric” pairs.) Consequently, under the bijection ▽m
Bn , the element w0 contains a
(w
;
w
,
.
.
.
,
w
type B cycle of length ℓ if and only if ▽m
0
1
m ) contains a zero block of size
Bn
mℓ.
The m-divisible non-crossing partitions of type Dn cannot be realised as certain
“partitions non croisées d’un cycle,” but as non-crossing partitions on an annulus with
2m(n − 1) vertices on the outer cycle and 2m vertices on the inner cycle, the vertices
on the outer cycle being labelled by 1, 2, . . . , mn − m, 1̄, 2̄, . . . , mn − m in clockwise
order, and the vertices of the inner cycle being labelled by mn − m + 1, . . . , mn − 1,
mn, mn − m + 1, . . . , mn − 1, mn in counter-clockwise order. Given a partition π of
{1, 2, . . . , mn, 1̄, 2̄, . . . , mn}, we represent it on this annulus in a manner analogous to
Kreweras’ graphical representation of his partitions; namely, we represent each block
of π by connecting the vertices labelled by the elements of the block by curves in
clockwise order, the important additional requirement being here that the curves must
be drawn in the interior of the annulus. If it is possible to draw the curves in such a
way that no two curves intersect, then the partition is called a non-crossing partition on
the (2m(n − 1), 2m)-annulus. Figure 10 shows a non-crossing partition on the (15, 6)annulus.
With this definition, the m-divisible non-crossing partitions of type Dn are in bijection
with non-crossing partitions π on the (2m(n − 1), 2m)-annulus, in which successive
elements of a block (successive in the circular order in the graphical representation of
the block) are in successive congruence classes modulo m, which have the property that
if B is a block of π then also B := {x̄ : x ∈ B} is a block of π, and which satisfy an
additional restriction concerning their zero block. Here again, a zero block is a block
B with B = B. The announced additional restriction says that a zero block can only
occur if it contains all the vertices of the inner cycle, that is, mn − m + 1, . . . , mn −
1, mn, mn − m + 1, . . . , mn − 1, mn, and at least two further elements from the outer
cycle. We denote this set of non-crossing partitions on the (2m(n − 1), 2m)-annulus
g m (Dn ). A non-crossing partition in NC
g m (Dn ) can only have at most one zero
by NC
block. Figures 10 and 11 give examples of 3-divisible non-crossing partitions of type
D6 , Figure 10 one without a zero block, while Figure 11 one with a zero block. Again,
it is clear that the condition that B is a block of the partition if and only if B is a block
translates into the condition that the geometric realisation of the partition is invariant
under rotation by 180◦ .
In order to clearly sort out the differences to the earlier combinatorial realisations of
m-divisible non-crossing partitions of types An−1 and Bn , we stress that for type Dn
there are three major features which are not present for the former types: (1) here we
consider non-crossing partitions on an annulus; (2) it is not sufficient to impose the
40
C. KRATTENTHALER AND T. W. MÜLLER
14
15
1
2
3
13
4
12
5
11
6
10
18
9
8
8
16
16
7
7
17
17
9
18
6
10
11
5
12
4
13
3
2
1
15
14
Figure 10. Combinatorial realisation of a 3-divisible non-crossing partition of type D6
condition that the size of every block is divisible by m: the condition on successive
elements of a block is stronger; (3) there is the above additional restriction on the zero
block (which is not present in type Bn ).
Given an element (w0 ; w1 , . . . , wm ) ∈ NC m (Dn ), the bijection, ▽m
Dn say, from [29]
works as follows. Recalling from Section 2 that W (Dn ) can be combinatorially realised
as a subgroup of the group of permutations of {1, 2, . . . , n, 1̄, 2̄, . . . , n̄}, and that, in
this realisation, the standard choice of a Coxeter element is c = [1, 2, . . . , n − 1] [n] =
(1, 2, . . . , n − 1, 1̄, 2̄, . . . , n − 1) (n, n̄), the announced bijection maps (w0 ; w1 , . . . , wm ) ∈
NC m (Dn ) to
▽m
Dn (w0 ; w1 , . . . , wm ) = [1, 2, . . . , m(n − 1)] [mn − m + 1, . . . , mn − 1, mn]
◦ (τ̄m,1 (w1 ))−1 (τ̄m,2 (w2 ))−1 · · · (τ̄m,m (wm ))−1 ,
where τ̄m,i is defined as above. We refer the reader to [29] for the details. For example,
let n = 6, m = 3, w0 = ((2, 4̄)), w1 = ((2, 6̄)) ((4, 5)), w2 = ((1, 5̄)) ((2, 3)), and
DECOMPOSITION NUMBERS FOR FINITE COXETER GROUPS
14
15
1
41
2
3
13
4
12
5
11
6
10
18
9
8
8
16
16
7
7
17
17
9
18
6
10
11
5
12
4
13
3
2
1
15
14
Figure 11. A 3-divisible non-crossing partition of type D6 with zero block
w3 = ((3, 6)). Then (w0 ; w1 , w2, w3 ) is mapped to
▽3D6 (w0 ;w1 , w2 , w3 )
= [1, 2, . . . , 15] [16, 17, 18]((4, 16)) ((10, 13)) ((2, 14)) ((5, 8)) ((9, 18))
= ((1, 2, 15)) ((3, 4, 17, 18, 10, 14)) ((5, 9, 16)) ((6, 7, 8)) ((11, 12, 13)).
(7.7)
Figure 10 shows the graphical representation of (7.7).
m
It is shown in [29] that ▽m
Dn is in fact an isomorphism between the posets NC (Dn )
g m (Dn ). Furthermore, it is proved in [29] that
and NC
ci (w0 ) = bi (▽m
Dn (w0 ; w1 , . . . , wm )),
i = 1, 2, . . . , n,
(7.8)
where ci (w0 ) denotes the number of type A cycles of length i of w0 and bi (π) denotes
one half of the number of non-zero blocks of size mi in the non-crossing partition
π. (Recall that non-zero blocks come in “symmetric” pairs.) Consequently, under
the bijection ▽m
Dn , the element w0 contains a type D cycle of length ℓ if and only if
m
▽Dn (w0 ; w1 , . . . , wm ) contains a zero block of size mℓ.
42
C. KRATTENTHALER AND T. W. MÜLLER
8. Decomposition numbers with free factors, and enumeration in the
poset of generalised non-crossing partitions
This section is devoted to applying our formulae from Sections 4–6 for the decomposition numbers of the types An , Bn , and Dn to the enumerative theory of generalised
non-crossing partitions for these types. Theorems 11–15 present formulae for the number of minimal factorisations of Coxeter elements in types An , Bn , and Dn , respectively,
where we do not prescribe the types of all the factors as for the decomposition numbers,
but just for some of them, while we impose rank sum conditions on other factors. Immediate corollaries are formulae for the number of multi-chains π1 ≤ π2 ≤ · · · ≤ πl−1 ,
g m (An−1 ), NC
g m (Bn ), and NC
g m (Dn ), where the poset
l being given, in the posets NC
rank of πi equals ri , and where the block structure of π1 is prescribed, see Corollaries 12,
14, and 16. These results in turn imply all known enumerative results on ordinary and
generalised non-crossing partitions via appropriate summations, see the remarks accompanying the corollaries. They also imply two further new results on chain enumeration
g m (Dn ), see Corollaries 18 and 19. We want to stress that, since NC
g m (Φ) and
in NC
m
NC (Φ) are isomorphic as posets for Φ = An−1 , Bn , Dn , Corollaries 12, 14, 16, 17, 18
g m (Φ), Φ = An−1 , Bn , Dn , via (7.4),
imply obvious results for NC m (Φ) in place of NC
(7.6), respectively (7.8).
We begin with our results for type An . The next theorem generalises Theorem 6,
which can be obtained from the former as the special case in which l = 1 and m1 = 1.
Theorem 11. For a positive integer d, let the types T1 , T2 , . . . , Td be given, where
(i)
(i)
m
(i)
m
n
Ti = A1 1 ∗ A2 2 ∗ · · · ∗ Am
n ,
i = 1, 2, . . . , d,
and let l, m1 , m2 , . . . , ml , s1 , s2 , . . . , sl be given non-negative integers with
rk T1 + rk T2 + · · · + rk Td + s1 + s2 + · · · + sl = n.
Then the number of factorisations
(1) (1)
(2) (2)
(l) (l)
(1)
(2)
(l)
c = σ1 σ2 · · · σd σ1 σ2 · · · σm
σ σ2 · · · σm
· · · σ1 σ2 · · · σm
,
1 1
2
l
(8.1)
where c is a Coxeter element in W (An ), such that the type of σi is Ti , i = 1, 2, . . . , d,
and such that
(i)
(i)
(i)
ℓT (σ1 ) + ℓT (σ2 ) + · · · + ℓT (σm
) = si ,
i
i = 1, 2, . . . , l,
(8.2)
is given by
(n + 1)d−1
d
Y
i=1
!
n − rk Ti + 1
1
(i)
(i)
n − rk Ti + 1 m(i)
1 , m2 , . . . , mn
ml (n + 1)
m1 (n + 1) m2 (n + 1)
, (8.3)
···
×
sl
s2
s1
where the multinomial coefficient is defined as in Lemma 4.
(j)
Proof. In the factorisation (8.1), we first fix also the types of the σi ’s. For i =
(j)
1, 2, . . . , mj and j = 1, 2, . . . , l, let the type of σi be
(j)
Ti
(i,j)
m
= A1 1
(i,j)
m
∗ A2 2
(i,j)
∗ · · · ∗ Anmn .
DECOMPOSITION NUMBERS FOR FINITE COXETER GROUPS
43
We know that the number of these factorisations is given by (4.1) with d replaced by
(j)
d+m1 +m2 +· · ·+ml and the appropriate interpretations of the mi ’s. Next we fix non(j)
(j)
negative integers ri and sum the expression (4.1) over all possible types Ti of rank
(j)
ri , i = 1, 2, . . . , mj , j = 1, 2, . . . , l. The corresponding summations are completely
analogous to the summation in the proof of Theorem 6. As a result, we obtain
!
d
Y
n
−
rk
T
+
1
1
i
(n + 1)d−1
(i)
(i)
n − rk Ti + 1 m1 , m2 , . . . , m(i)
n
i=1
n+1
n+1 n+1
n+1
n+1 n+1
···
×
···
×
(2)
(2)
(2)
(1)
(1)
(1)
rm2
r2
r1
rm1
r2
r1
n+1
n+1 n+1
···
×···×
(l)
(l)
(l)
rml
r2
r1
for the number of factorisations under consideration. In view of (8.2) and (2.4), to
obtain the final result, we must sum these expressions over all non-negative integers
(1)
(l)
r1 , . . . , rml satisfying the equations
(j)
(j)
(j)
r1 + r2 + · · · + rm
= sj ,
j
j = 1, 2, . . . , l.
(8.4)
This is easily done by means of the multivariate version of the Chu–Vandermonde
summation. The formula in (8.3) follows.
In view of the combinatorial realisation of m-divisible non-crossing partitions of type
An−1 which we described in Section 7, the special case d = 1 of the above theorem has
the following enumerative consequence.
Corollary 12. Let l be a positive integer, and let s1 , s2 , . . . , sl be non-negative integers
with s1 + s2 + · · · + sl = n − 1. The number of multi-chains π1 ≤ π2 ≤ · · · ≤ πl−1 in the
g m (An−1 ), with the property that rk(πi ) = s1 + s2 + · · · + si , i = 1, 2, . . . , l − 1,
poset NC
and that the number of blocks of size mi of π1 is bi , i = 1, 2, . . . , n, is given by
1
mn
mn
b1 + b2 + · · · + bn
,
(8.5)
···
sl
s2
b1 , b2 , . . . , bn
b1 + b2 + · · · + bn
provided that b1 + 2b2 + · · · + nbn ≤ n, and is 0 otherwise.
Remark. The conditions in the statement of the corollary imply that
s1 + b1 + b2 + · · · + bn = n.
(8.6)
π1 ≤ π2 ≤ · · · ≤ πl−1
(8.7)
Proof. Let
g m (An−1 ). Suppose that, under the bijection ▽m , the element
be a multi-chain in NC
An−1
(j)
(j)
(j)
πj corresponds to the tuple (w0 ; w1 , . . . , wm ), j = 1, 2, . . . , l − 1. The inequalities in
(1)
(1)
(1)
(8.7) imply that w1 , w2 , . . . , wm can be factored in the form
(1)
wi
(l)
(l−1)
where ui = wi
(j)
wi
(2) (3)
(l)
= ui ui · · · ui ,
i = 1, 2, . . . , m,
and, more generally,
(j+1) (j+2)
(l)
ui
· · · ui ,
= ui
i = 1, 2, . . . , m, j = 1, 2, . . . , l − 1.
(8.8)
44
C. KRATTENTHALER AND T. W. MÜLLER
For later use, we record that
(j)
(j)
(j)
c = w0 w1 · · · wm
(j)
= w0
(j+1) (j+2)
u1
u1
(l) · · · u1
(j+1) (j+2)
(l) u2
· · · u2 · · ·
u2
(l)
u(j+1)
u(j+2)
· · · um
.
m
m
(8.9)
Now, by (7.4), the block structure conditions on π1 in the statement of the corollary
(1)
translate into the condition that the type of w0 is
n
Ab12 ∗ Ab23 ∗ · · · ∗ Abn−1
.
(8.10)
On the other hand, using (7.2), we see that the rank conditions in the statement of the
corollary mean that
(j)
ℓT (w0 ) = s1 + s2 + · · · + sj ,
j = 1, 2, . . . , l − 1.
In combination with (8.9), this yields the conditions
(j)
(j)
ℓT (u1 ) + ℓT (u2 ) + · · · + ℓT (u(j)
m ) = sj ,
j = 2, 3, . . . , l.
Thus, we want to count the number of factorisations
(1)
(2) (3)
(l) (2) (3)
(l) (3)
(l)
c = w0 u1 u1 · · · u1 u2 u2 · · · u2 · · · u(2)
m um · · · um ,
(8.11)
(8.12)
(1)
where the type of w0 is given in (8.10), and where the “rank conditions” (8.11) are
satisfied. So, in view of (2.4), we are in the situation of Theorem 11 with n replaced by
n − 1, d = 1, l replaced by l − 1, si replaced by si+1 , i = 1, 2, . . . , l − 1, T1 the type in
(8.10), m1 = m2 = · · · = ml−1 = m, except that the factors are not exactly in the order
as in (8.1). However, by (2.2) we know that the order of factors is without relevance.
Therefore we just have to apply Theorem 11 with the above specialisations. If we also
take into account (8.6), then we arrive immediately at (8.5).
This result is new even for m = 1, that is, for the poset of Kreweras’ non-crossing partitions of {1, 2, . . . , n}. It implies all known results on Kreweras’ non-crossing partitions
and the m-divisible non-crossing partitions of Edelman. Namely, for l = 2 it reduces
to Armstrong’s result [1, Theorem 4.4.4 with ℓ = 1] on the number of m-divisible
g m (An−1 ) with a given block structure, which itself connon-crossing partitions in NC
tains Kreweras’ result [30, Theorem 4] on his non-crossing partitions with a given block
structure as a special case. If we sum the expression (8.5) over all s2 , s3 , . . . , sl with
s2 + s3 + · · · + sl = n − 1 − s1 , then we obtain that the number of all multi-chains
g m (An−1 ) of m-divisible non-crossing
π1 ≤ π2 ≤ · · · ≤ πl−1 in Edelman’s poset NC
partitions of {1, 2, . . . , mn} in which π1 has bi blocks of size mi equals
1
(l − 1)mn
b1 + b2 + · · · + bn
n − s1 − 1
b1 , b2 , . . . , bn
b1 + b2 + · · · + bn
1
(l − 1)mn
b1 + b2 + · · · + bn
=
, (8.13)
b1 + b2 + · · · + bn − 1
b1 , b2 , . . . , bn
b1 + b2 + · · · + bn
provided that b1 + 2b2 + · · · + nbn ≤ n, a result originally due to Armstrong [1,
Theorem 4.4.4]. On the other hand, if we sum the expression (8.5) over all possible
b1 , b2 , . . . , bn , that is, b2 + 2b3 + · · · + (n − 1)bn = s1 , use of Lemma 4 with M = n − s1
and r = s1 yields that the number of all multi-chains π1 ≤ π2 ≤ · · · ≤ πl−1 in Edelman’s
DECOMPOSITION NUMBERS FOR FINITE COXETER GROUPS
45
g m (An−1 ) ∼
poset NC
= NC m (An−1 ) where πi is of rank s1 + s2 + · · ·+ si , i = 1, 2, . . . , l −1,
equals
1 n
mn
mn
,
(8.14)
···
sl
s2
n s1
a result originally due to Edelman [18, Theorem 4.2]. Clearly, this formula contains
at the same time a formula for the number of all m-divisible non-crossing partitions of
{1, 2, . . . , mn} with a given number of blocks upon setting l = 2 (cf. [18, Lemma 4.1]),
as well as that it implies that the total number of multi-chains π1 ≤ π2 ≤ · · · ≤ πl−1 in
the poset of these partitions is
1 (l − 1)mn + n
(8.15)
n−1
n
upon summing (8.14) over all non-negative integers s1 , s2 , . . . , sl with s1 + s2 + · · ·+ sl =
n − 1 by means of the multivariate Chu–Vandermonde summation, thus recovering the
formula [18, Cor. 4.4] for the zeta polynomial of the poset of m-divisible non-crossing
partitions of type An−1 . As special case l = 2, we recover the well-known fact that
the
1 (m+1)n
total number of m-divisible non-crossing partitions of {1, 2, . . . , mn} is n n−1 .
We continue with our results for type Bn . We formulate the theorem below on
factorisations in W (Bn ) only with restrictions on the combinatorial type of some factors.
An analogous result with group-theoretical type instead could be easily derived as well.
We omit this here because, for the combinatorial applications that we have in mind,
combinatorial type suffices. We remark that the theorem generalises Theorem 8, which
can be obtained from the former as the special case in which l = 1 and m1 = 1.
Theorem 13. (i) For a positive integer d, let the types T1 , T2 , . . . , Td be given, where
(i)
m
(i)
(i)
m
Ti = A1 1 ∗ A2 2 ∗ · · · ∗ Anmn ,
and
(j)
m
i = 1, 2, . . . , j − 1, j + 1, . . . , d,
(j)
(j)
m
n
Tj = Bα ∗ A1 1 ∗ A2 2 ∗ · · · ∗ Am
,
n
for some α ≥ 1, and let l, m1 , m2 , . . . , ml , s1 , s2 , . . . , sl be given non-negative integers
with
rk T1 + rk T2 + · · · + rk Td + s1 + s2 + · · · + sl = n.
(8.16)
Then the number of factorisations
(1) (1)
(2) (2)
(l) (l)
(1)
(2)
(l)
c = σ1 σ2 · · · σd σ1 σ2 · · · σm
σ σ2 · · · σm
· · · σ1 σ2 · · · σm
,
1 1
2
l
(8.17)
where c is a Coxeter element in W (Bn ), such that the combinatorial type of σi is Ti ,
i = 1, 2, . . . , d, and such that
(i)
(i)
(i)
ℓT (σ1 ) + ℓT (σ2 ) + · · · + ℓT (σm
) = si ,
i
i = 1, 2, . . . , l,
is given by
Y
!
d
n
−
rk
T
n
−
rk
T
1
j
i
nd−1
(i)
(i)
(i)
(j)
(j)
(j)
n
−
rk
T
m1 , m2 , . . . , mn
i m1 , m2 , . . . , mn
i=1
i6=j
(8.18)
ml n
m1 n m2 n
, (8.19)
···
×
sl
s2
s1
46
C. KRATTENTHALER AND T. W. MÜLLER
where the multinomial coefficient is defined as in Lemma 4.
(ii) For a positive integer d, let the types T1 , T2 , . . . , Td be given, where
(i)
(i)
m
(i)
m
n
Ti = A1 1 ∗ A2 2 ∗ · · · ∗ Am
n ,
i = 1, 2, . . . , d,
and let l, m1 , m2 , . . . , ml , s1 , s2 , . . . , sl be given non-negative integers. Then the number
of factorisations (8.17) which satisfy (8.18) plus the condition that the combinatorial
type of σi is Ti , i = 1, 2, . . . , d, is given by
!
d
Y
n
−
rk
T
1
i
nd−1 (n − rk T1 − rk T2 − · · · − rk Td )
(i)
(i)
n − rk Ti m(i)
1 , m2 , . . . , mn
i=1
ml n
m1 n m2 n
. (8.20)
···
×
sl
s2
s1
Proof. We start with the proof of item (i). In the factorisation (8.17), we first fix also
(j)
(j)
the types of the σi ’s. For i = 1, 2, . . . , mj and j = 1, 2, . . . , l let the type of σi be
(j)
Ti
(i,j)
m
= A1 1
(i,j)
m
∗ A2 2
(i,j)
∗ · · · ∗ Anmn .
We know that the number of these factorisations is given by (5.1) with d replaced by
(j)
d+m1 +m2 +· · ·+ml and the appropriate interpretations of the mi ’s. Next we fix non(j)
(j)
negative integers ri and sum the expression (5.1) over all possible types Ti of rank
(j)
ri , i = 1, 2, . . . , mj , j = 1, 2, . . . , l. The corresponding summations are completely
analogous to the first summation in the proof of Theorem 8. As a result, we obtain
!
Y
d
n
−
rk
T
n
−
rk
T
1
j
i
nd−1
(i)
(i)
(i)
(j)
(j)
(j)
n
−
rk
T
m1 , m2 , . . . , mn
i m1 , m2 , . . . , mn
i=1
n
× (1)
r1
i6=j
n
n
n
n
n
× (2)
(2)
(2) · · ·
(1)
(1) · · ·
rm2
r2
r1
rm1
r2
n
n
n
× · · · × (l)
(l)
(l) · · ·
rml
r2
r1
for the number of factorisations under consideration. In view of (8.18) and (2.4), to
obtain the final result, we must sum these expressions over all non-negative integers
(1)
(l)
r1 , . . . , rml satisfying the equations
(j)
(j)
(j)
r1 + r2 + · · · + rm
= sj ,
j
j = 1, 2, . . . , l.
This is easily done by means of the multivariate version of the Chu–Vandermonde
summation. The formula in (8.19) follows.
The proof of item (ii) is completely analogous, we must, however, cope with the
complication that the type B cycle, which, according to Theorem 7, must occur in
the disjoint cycle decomposition of exactly one of the factors on the right-hand side of
(j)
(j)
(8.17), can occur in any of the σi ’s. So, let us fix the types of the σi ’s to
(j)
Ti
(i,j)
m
= A1 1
(i,j)
m
∗ A2 2
(i,j)
∗ · · · ∗ Anmn ,
DECOMPOSITION NUMBERS FOR FINITE COXETER GROUPS
47
i = 1, 2, . . . , mj , j = 1, 2, . . . , l, except for (i, j) = (p, q), where we require that the type
(q)
of σp is
e1
e2
en
Tp(q) = Bα ∗ A1m
∗ A2m
∗ · · · ∗ Am
n .
Again, we know that the number of these factorisations is given by (5.1) with d replaced
(j)
by d + m1 + m2 + · · · + ml and the appropriate interpretations of the mi ’s. Now we
(j)
(j)
fix non-negative integers ri and sum the expression (5.1) over all possible types Ti
(j)
of rank ri , i = 1, 2, . . . , mj , j = 1, 2, . . . , l. Again, the corresponding summations are
completely analogous to the summations in the proof of Theorem 8. In particular, the
(q)
(q)
summation over all possible types Tp of rank rp is essentially the summation on the
right-hand side of (5.14) with d replaced by d + m1 + m2 + · · · + ml and r replaced
(q)
by rp . If we use what we know from the proof of Theorem 8, then the result of the
summations is found to be
!
d
Y
n
−
rk
T
1
i
nd−1
(i)
(i)
n − rk Ti m1 , m2 , . . . , m(i)
n
i=1
n
n
n
n
n
n
(q)
× · · · × (q) · · · rp
× (1)
(q)
(q) · · ·
(1)
(1) · · ·
rmq
rp
r1
rm1
r2
r1
n
n
n
× · · · × (l)
(l) . (8.21)
(l) · · ·
rml
r2
r1
(q) n The reader should note that the term rp rp(q)
in this expression results from the sum(q)
(q)
(q)
mation over all types Tp of rank rp (compare (5.15) with r replaced by rp ; we have
(q)
rp
(q)
n−1 n =
over
(q) ). Using (8.16), (8.18) and (2.4), we see that the sum of all rp
(q)
n rp
rp −1
p = 1, 2, . . . , mq and q = 1, 2, . . . , l must be n − rk T1 − rk T2 − · · · − rk Td . Hence, the
sum of the expressions (8.21) over all (p, q) equals
!
d
Y
n
−
rk
T
1
i
nd−1 (n − rk T1 − rk T2 − · · · − rk Td )
(i)
(i)
n − rk Ti m1 , m2 , . . . , m(i)
n
i=1
n
n
n
n
n
n
× · · · × (q)
× (1)
(q)
(q) · · ·
(1)
(1) · · ·
rmq
r2
r1
rm1
r2
r1
n
n
n
× · · · × (l)
(l) .
(l) · · ·
rml
r2
r1
(1)
(l)
Finally, we must sum these expressions over all non-negative integers r1 , . . . , rml satisfying the equations
(j)
(j)
(j)
r1 + r2 + · · · + rm
= sj ,
j
j = 1, 2, . . . , l.
Once again, this is easily done by means of the multivariate version of the Chu–
Vandermonde summation. As a result, we obtain the formula in (8.20).
In view of the combinatorial realisation of m-divisible non-crossing partitions of type
Bn which we described in Section 7, the special case d = 1 of the above theorem has
the following enumerative consequence.
48
C. KRATTENTHALER AND T. W. MÜLLER
Corollary 14. Let l be a positive integer, and let s1 , s2 , . . . , sl be non-negative integers
with s1 + s2 + · · · + sl = n. The number of multi-chains π1 ≤ π2 ≤ · · · ≤ πl−1 in the
g m (Bn ) with the property that rk(πi ) = s1 + s2 + · · · + si , i = 1, 2, . . . , l − 1, and
poset NC
that the number of non-zero blocks of π1 of size mi is 2bi , i = 1, 2, . . . , n, is given by
mn
mn
b1 + b2 + · · · + bn
,
(8.22)
···
sl
s2
b1 , b2 , . . . , bn
provided that b1 + 2b2 + · · · + nbn ≤ n, and is 0 otherwise.
Remark. The conditions in the statement of the corollary imply that
s1 + b1 + b2 + · · · + bn = n.
(8.23)
g m (Bn )
The reader should recall from Section 7, that non-zero blocks of elements π of NC
occur in pairs since, with a block B of π, also B is a block of π.
Proof. The arguments are completely analogous to those of the proof of Corollary 12.
The conclusion here is that we need Theorem 13 with d = 1, l replaced by l − 1, si
replaced by si+1 , i = 1, 2, . . . , l − 1, m1 = m2 = · · · = ml−1 = m, and T1 of the type
n
Bn−b1 −2b2 −···−nbn ∗ Ab12 ∗ Ab23 ∗ · · · ∗ Abn−1
in the case that b1 + 2b2 + · · · + nbn < n (which enforces the existence of a zero block
of size 2(n − b1 − 2b2 − · · · − nbn ) in π1 ), respectively
n
Ab12 ∗ Ab23 ∗ · · · ∗ Abn−1
if not. So, depending on the case in which we are, we have to apply (8.19), respectively
(8.20). However, for d = 1 these two formulae become identical. More precisely, under
the above specialisations, they reduce to
mn
mn
n − rk T1
.
···
sl
s2
b2 , b3 , . . . , bn
If we also take into account (8.23), then we arrive immediately at (8.22).
This result is new even for m = 1, that is, for the poset of Reiner’s type Bn
non-crossing partitions. It implies all known results on these non-crossing partitions
and their extension to m-divisible type Bn non-crossing partitions due to Armstrong.
Namely, for l = 2 it reduces to Armstrong’s result [1, Theorem 4.5.11 with ℓ = 1] on
g m (Bn ) with a given block structure, which itself conthe number of elements of NC
tains Athanasiadis’ result [2, Theorem 2.3] on Reiner’s type Bn non-crossing partitions
with a given block structure as a special case. If we sum the expression (8.22) over all
s2 , s3 , . . . , sl with s2 + s3 + · · · + sl = n − s1 , then we obtain that the number of all
g m (Bn ) in which π1 has 2bi non-zero blocks of
multi-chains π1 ≤ π2 ≤ · · · ≤ πl−1 in NC
size mi equals
(l − 1)mn
b1 + b2 + · · · + bn
(l − 1)mn
b1 + b2 + · · · + bn
,
=
b1 + b2 + · · · + bn
b1 , b2 , . . . , bn
n − s1
b1 , b2 , . . . , bn
(8.24)
provided that b1 + 2b2 + · · · + nbn ≤ n, a result originally due to Armstrong [1, Theorem 4.5.11]. On the other hand, if we sum the expression (8.22) over all possible
b1 , b2 , . . . , bn , that is, over b2 + 2b3 + · · · + (n − 1)bn ≤ s1 , use of Lemma 4 with
M = n − s1 and r = s1 − α (where α stands for the difference between s1 and
DECOMPOSITION NUMBERS FOR FINITE COXETER GROUPS
49
b2 + 2b3 + · · ·+ (n−1)bn ) yields that the number of all multi-chains π1 ≤ π2 ≤ · · · ≤ πl−1
g m (Bn ) ∼
in NC
= NC m (Bn ) where πi is of rank s1 + s2 + · · · + si, i = 1, 2, . . . , l − 1, equals
n X
mn
mn
n
mn
n − α − 1 mn
,
(8.25)
···
=
···
sl
s2
s1
sl
s2
s1 − α
α=0
another result due to Armstrong [1, Theorem 4.5.7]. Clearly, this formula contains
g m (Bn ) ∼
at the same time a formula for the number of all elements of NC
= NC m (Bn )
with a given number of blocks (equivalently, a given rank) upon setting l = 2 (cf.
[1, Theorem 4.5.8]), as well as that it implies that the total number of multi-chains
g m (Bn ) ∼
π1 ≤ π2 ≤ · · · ≤ πl−1 in NC
= NC m (Bn ) is
(l − 1)mn + n
(8.26)
n
upon summing (8.25) over all non-negative integers s1 , s2 , . . . , sl with s1 + s2 + · · ·+ sl =
n by means of the multivariate Chu–Vandermonde summation, thus recovering the
formula [1, Theorem 3.6.9] for the zeta polynomial of the poset of generalised noncrossing partitions in the case of type Bn . As special case
l = 2, we recover the fact
(m+1)n
m
m
∼
g
that the cardinality of NC (Bn ) = NC (Bn ) is
(cf. [1, Theorem 3.5.3]).
n
The final set of results in this section concerns the type Dn . We start with Theorem 15, the result on factorisations in W (Dn ) which is analogous to Theorems 11 and
13. Similar to Theorem 13, we formulate the theorem only with restrictions on the
combinatorial type of some factors. An analogous result with group-theoretical type
instead could be easily derived as well. We refrain from doing this here because, again,
for the combinatorial applications that we have in mind, combinatorial type suffices.
We remark that the theorem generalises Theorem 10, which can be obtained from the
former as the special case in which l = 1 and m1 = 1.
Theorem 15. (i) For a positive integer d, let the types T1 , T2 , . . . , Td be given, where
(i)
m
(i)
(i)
m
n
Ti = A1 1 ∗ A2 2 ∗ · · · ∗ Am
n ,
i = 1, 2, . . . , j − 1, j + 1, . . . , d,
and
(j)
m
(j)
m
(j)
n
Tj = Dα ∗ A1 1 ∗ A2 2 ∗ · · · ∗ Am
,
n
for some α ≥ 2, and let l, m1 , m2 , . . . , ml , s1 , s2 , . . . , sl be given non-negative integers
with
rk T1 + rk T2 + · · · + rk Td + s1 + s2 + · · · + sl = n.
Then the number of factorisations
(1) (1)
(2) (2)
(l) (l)
(1)
(2)
(l)
c = σ1 σ2 · · · σd σ1 σ2 · · · σm
σ σ2 · · · σm
· · · σ1 σ2 · · · σm
,
1 1
2
l
(8.27)
where c is a Coxeter element in W (Dn ), such that the combinatorial type of σi is Ti ,
i = 1, 2, . . . , d, and such that
(i)
(i)
(i)
ℓT (σ1 ) + ℓT (σ2 ) + · · · + ℓT (σm
) = si ,
i
i = 1, 2, . . . , l,
(8.28)
50
C. KRATTENTHALER AND T. W. MÜLLER
is given by
(n − 1)d−1
n − rk Tj
(j)
(j)
(j)
m1 , m2 , . . . , mn
!
n − rk Ti − 1
1
(i)
(i)
n − rk Ti − 1 m(i)
1 , m2 , . . . , mn
i=1
Y
d
i6=j
ml (n − 1)
m1 (n − 1) m2 (n − 1)
, (8.29)
···
×
sl
s2
s1
the multinomial coefficient being defined as in Lemma 4.
(ii) For a positive integer d, let the types T1 , T2 , . . . , Td be given, where
(i)
m
(i)
(i)
m
n
Ti = A1 1 ∗ A2 2 ∗ · · · ∗ Am
n ,
i = 1, 2, . . . , d,
and let l, m1 , m2 , . . . , ml , s1 , s2 , . . . , sl be given non-negative integers. Then the number
of factorisations (8.27) which satisfy (8.28) as well as the condition that the combinatorial type of σi is Ti , i = 1, 2, . . . , d, is given by
2(n − 1)d−1
d X
j=1
n − rk Tj
(j)
(j)
(j)
m1 , m2 , . . . , mn
!
n − rk Ti − 1
1
(i)
(i)
n − rk Ti − 1 m(i)
1 , m2 , . . . , mn
i=1
Y
d
i6=j
ml (n − 1)
m1 (n − 1) m2 (n − 1)
···
×
sl
s2
s1
!
d
Y
n − rk Ti − 1
1
+ (n − 1)d
(i)
(i)
n − rk Ti − 1 m1 , m(i)
2 , . . . , mn
i=1
l
X
ml (n − 1)
mj (n − 1) − 1
m1 (n − 1)
···
···
×
mj
sl
s
−
2
s
j
1
j=1
!
d
Y
n
−
rk
T
−
1
1
i
− 2(d − 1)(n − 1)d
(i)
(i)
n − rk Ti − 1 m(i)
1 , m2 , . . . , mn
i=1
ml (n − 1)
m1 (n − 1) m2 (n − 1)
. (8.30)
···
×
sl
s2
s1
Proof. The proof of item (i) is completely analogous to the proof of item (i) in Theorem 13. Making reference to that proof, the only difference is that, instead of the
expression (5.1), we must use (6.1) with d replaced by d + m1 + m2 + · · · + ml and
(j)
(j)
the appropriate interpretations of the mi ’s. The summations over types Ti with
(j)
fixed rank ri are carried out by using (3.4) with M = n − r − 1. Subsequently, the
(j)
summations over the ri ’s satisfying (8.4) are done by the multivariate version of the
Chu–Vandermonde summation. We leave it to the reader to fill in the details to finally
arrive at (8.29).
Similarly, the proof of item (ii) is analogous to the proof of item (ii) in Theorem 13.
However, we must cope with the complication that there may or may not be a type D
(j)
cycle in the disjoint cycle decomposition of one of the σi ’s on the right-hand side of
DECOMPOSITION NUMBERS FOR FINITE COXETER GROUPS
51
(j)
(8.27). In the case that there is no type B cycle, we fix the types of the σi ’s to
(j)
Ti
(i,j)
m
(i,j)
m
∗ A2 2
= A1 1
(i,j)
∗ · · · ∗ Anmn ,
i = 1, 2, . . . , mj , j = 1, 2, . . . , l, and sum the expression (6.2) with d replaced by d +
(j)
m1 + m2 + · · · + ml and the appropriate interpretations of the mi ’s over all possible
(j)
(j)
types Ti with rank ri , i = 1, 2, . . . , mj , j = 1, 2, . . . , l. This yields the expression
2(n − 1)d−1
d X
j=1
n − rk Tj
(j)
(j)
(j)
m1 , m2 , . . . , mn
Y
d
n − rk Ti − 1
1
(i)
(i)
n − rk Ti − 1 m1 , m(i)
2 , . . . , mn
i=1
i6=j
!
n−1
n−1 n−1
n−1
n−1 n−1
···
×···×
···
×
(l)
(l)
(l)
(1)
(1)
(1)
rml
r2
r1
rm1
r2
r1
!
mj d
l X
Y
X
n
−
rk
T
−
1
1
i
+ 2(n − 1)d
(i)
(i)
n − rk Ti − 1 m(i)
1 , m2 , . . . , mn
j=1 i=1 i=1
n−1
n−1 n−1
n−1
n−1 n−1
···
×···×
···
×
(l)
(l)
(l)
(1)
(1)
(1)
rml
r2
r1
rm1
r2
r1
!
d
Y
n − rk Ti − 1
1
d−1
− 2(d − 1)(n − 1)
(i)
(i)
n − rk Ti − 1 m(i)
1 , m2 , . . . , mn
i=1
n−1
n−1 n−1
n−1
n−1 n−1
. (8.31)
···
×···×
···
×
(l)
(l)
(l)
(1)
(1)
(1)
rml
r2
r1
rm1
r2
r1
(q)
In the case that there appears, however, a type B cycle in σp , say, we adopt the same
(q)
set-up as above, except that we restrict σp to types of the form
e1
e2
m
en
Tp(q) = Dα ∗ A1m
∗ Am
2 ∗ · · · ∗ An .
Subsequently, we sum the expression (6.1) with d replaced by d + m1 + m2 + · · · + ml
(j)
(j)
(j)
and the appropriate interpretations of the mi ’s over all possible types Ti of rank ri .
This time, we obtain
!
n
−
rk
T
−
1
1
i
(n − 1)d
(i)
(i)
(i)
n
−
rk
T
−
1
m
,
m
,
.
. . , mn
i
1
2
q=1 p=1 α=2 i=1
n−1
n−α−1
n−1
n−1
n−1 n−1
···
···
×···×
···
×
(q)
(q)
(q)
(1)
(1)
(1)
rmq
rp − α
r1
rm1
r2
r1
n−1
n−1 n−1
. (8.32)
···
×···×
(l)
(l)
(l)
rml
r2
r1
mq n
d
l X
XY
X
The sum over α can be evaluated by means of the elementary summation formula
X
n n X
n−2
n−2
n−α−1
n−α−1
.
=
=
=
r
−
2
n
−
r
n
−
r
−
1
r
−
α
α=2
α=2
52
C. KRATTENTHALER AND T. W. MÜLLER
Finally, we must sum the expressions (8.31) and (8.32) over all non-negative integers
(1)
(l)
r1 , . . . , rml satisfying the equations
(j)
(j)
(j)
r1 + r2 + · · · + rm
= sj ,
j
j = 1, 2, . . . , l.
Once again, this is easily done by means of the multivariate version of the Chu–
Vandermonde summation. After some simplification, we obtain the formula in (8.30).
In view of the combinatorial realisation of m-divisible non-crossing partitions of type
Dn which we described in Section 7, the special case d = 1 of the above theorem has
the following enumerative consequence.
Corollary 16. Let l be a positive integer, and let s1 , s2 , . . . , sl be non-negative integers
with s1 + s2 + · · · + sl = n. The number of multi-chains π1 ≤ π2 ≤ · · · ≤ πl−1 in the
g m (Dn ) with the property that rk(πi ) = s1 + s2 + · · · + si , i = 1, 2, . . . , l − 1, and
poset NC
that the number of non-zero blocks of π1 of size mi is 2bi , i = 1, 2, . . . , n, is given by
m(n − 1)
m(n − 1)
b1 + b2 + · · · + bn
,
(8.33)
···
sl
s2
b1 , b2 , . . . , bn
if b1 + 2b2 + · · · + nbn < n − 1, and
m(n − 1)
m(n − 1)
b1 + b2 + · · · + bn
···
2
sl
s2
b1 , b2 , . . . , bn
m(n − 1)
b1 + b2 + · · · + bn − 1
+
b1 − 1, b2 , . . . , bn
b1 + b2 + · · · + bn − 1
l X
m(n − 1)
m(n − 1) − 1
m(n − 1)
, (8.34)
···
···
×
sl
sj − 2
s2
j=2
if b1 + 2b2 + · · · + nbn = n.
Remark. The conditions in the statement of the corollary imply that
s1 + b1 + b2 + · · · + bn = n.
(8.35)
g m (Dn )
The reader should recall from Section 7, that non-zero blocks of elements π of NC
occur in pairs since, with a block B of π, also B is a block of π. The condition
b1 + 2b2 + · · · + nbn < n − 1, which is required for Formula (8.33) to hold, implies that
π1 must contain a zero block of size 2(n − b1 − 2b2 − · · · − nbn ), while the equality
b1 + 2b2 + · · · + nbn = n, which is required for Formula (8.34) to hold, implies that
π1 contains no zero block. The extra condition on zero blocks that are imposed on
g m (Dn ) implies that b1 + 2b2 + · · · + nbn cannot be equal to n − 1.
elements of NC
Proof. Again, the arguments are completely analogous to those of the proof of Corollary 12. Here we need Theorem 15 with d = 1, l replaced by l − 1, si replaced by si+1 ,
i = 1, 2, . . . , l − 1, m1 = m2 = · · · = ml−1 = m, and T1 of the type
n
Dn−b1 −2b2 −···−nbn ∗ Ab12 ∗ Ab23 ∗ · · · ∗ Abn−1
in the case that b1 + 2b2 + · · · + nbn < n − 1, respectively
n
Ab12 ∗ Ab23 ∗ · · · ∗ Abn−1
DECOMPOSITION NUMBERS FOR FINITE COXETER GROUPS
53
if not. So, depending on the case in which we are, we have to apply (8.29), respectively
(8.30). If we also take into account (8.35), then we arrive at the claimed result after
little manipulation. Since we have done similar calculations already several times, the
details are left to the reader.
This result is new even for m = 1, that is, for the poset of type Dn non-crossing
partitions of Athanasiadis and Reiner [4], and of Bessis and Corran [9]. Not only does
it imply all known results on these non-crossing partitions and their extension to mdivisible type Dn non-crossing partitions due to Armstrong, it allows us as well to solve
several open enumeration problems on the m-divisible type Dn non-crossing partitions.
We state these new results separately in the corollaries below.
To begin with, if we set l = 2 in Corollary 16, then we obtain the following extension
g m (Dn ) of Athanasiadis and Reiner’s result [4, Theorem 1.3] on the number of
to NC
type Dn non-crossing partitions with a given block structure.
g m (Dn ) which have 2bi non-zero blocks
Corollary 17. The number of all elements of NC
of size mi equals
m(n − 1)
b1 + b2 + · · · + bn
(8.36)
b1 + b2 + · · · + bn
b1 , b2 , . . . , bn
if b1 + 2b2 + · · · + nbn < n − 1, and
m(n − 1)
b1 + b2 + · · · + bn
2
b1 + b2 + · · · + bn
b1 , b2 , . . . , bn
m(n − 1)
b1 + b2 + · · · + bn − 1
(8.37)
+
b1 + b2 + · · · + bn − 1
b1 − 1, b2 , . . . , bn
if b1 + 2b2 + · · · + nbn = n.
On the other hand, if we sum the expression (8.33), respectively (8.34), over all
s2 , s3 , . . . , sl with s2 + s3 + · · · + sl = n − s1 , then we obtain the following generalisation.
g m (Dn ) in
Corollary 18. The number of all multi-chains π1 ≤ π2 ≤ · · · ≤ πl−1 in NC
which π1 has 2bi non-zero blocks of size mi equals
(l − 1)m(n − 1)
b1 + b2 + · · · + bn
,
(8.38)
b1 + b2 + · · · + bn
b1 , b2 , . . . , bn
if b1 + 2b2 + · · · + nbn < n − 1, and
(l − 1)m(n − 1)
b1 + b2 + · · · + bn
2
b1 + b2 + · · · + bn
b1 , b2 , . . . , bn
(l − 1)m(n − 1)
b1 + b2 + · · · + bn − 1
(8.39)
+
b1 + b2 + · · · + bn − 1
b1 − 1, b2 , . . . , bn
if b1 + 2b2 + · · · + nbn = n.
Next we sum the expressions (8.33) and (8.34) over all possible b1 , b2 , . . . , bn , that is,
we sum (8.33) over b2 + 2b3 + · · · + (n − 1)bn < s1 − 1, and we sum the expression (8.34)
over b2 + 2b3 + · · · + (n − 1)bn = s1 . With the help of Lemma 4 and the simple binomial
summation (6.24), these sums can indeed be evaluated. In this manner, we obtain the
g m (Dn ).
following result on rank-selected chain enumeration in NC
54
C. KRATTENTHALER AND T. W. MÜLLER
g m (Dn ) ∼
Corollary 19. The number of all multi-chains π1 ≤ π2 ≤ · · · ≤ πl−1 in NC
=
m
NC (Dn ) where πi is of rank s1 + s2 + · · · + si , i = 1, 2, . . . , l − 1, equals
m(n − 1)
n − 1 m(n − 1)
···
2
sl
s2
s1
l
X
m(n − 1)
m(n − 1) − 1
n − 1 m(n − 1)
···
···
+m
sl
sj − 2
s2
s1
j=2
m(n − 1)
m(n − 1)
n−2
. (8.40)
···
+
sl
s2
s1 − 2
This formula extends Athanasiadis and Reiner’s formula [4, Theorem 1.2(ii)] from
g m (Dn ). Setting l = 2, we obtain a formula for the number of all
NC(Dn ) to NC
g m (Dn ) ∼
elements in NC
= NC m (Dn ) with a given number of blocks (equivalently, of
given rank); cf. [1, Theorem 4.6.3]. Next, summing (8.40) over all non-negative integers
s1 , s2 , . . . , sl with s1 + s2 + · · · + sl = n by means of the multivariate Chu–Vandermonde
summation, we find that the total number of multi-chains π1 ≤ π2 ≤ · · · ≤ πl−1 in
g m (Dn ) ∼
NC
= NC m (Dn ) is given by
((l − 1)m + 1)(n − 1)
((l − 1)m + 1)(n − 1)
+
2
n−1
n
2(l − 1)m(n − 1) + n ((l − 1)m + 1)(n − 1)
, (8.41)
=
n−1
n
thus recovering the formula [1, Theorem 3.6.9] for the zeta polynomial of the poset
of generalised non-crossing partitions for the type Dn . The special case l = 2 of
g m (Dn ) ∼
(8.41) gives the well-known fact that the cardinality of NC
= NC m (Dn ) is
2m(n−1)+n (m+1)(n−1)
(cf. [1, Theorem 3.5.3]).
n
n−1
In the following section, Corollary 19 will enable us to provide a new proof of Armstrong’s F = M (Ex-)Conjecture in type Dn .
9. Proof of the F = M Conjecture for type D
Armstrong’s F = M (Ex-)Conjecture [1, Conjecture 5.3.2], which extends an earlier
conjecture of Chapoton [17], relates the “F -triangle” of the generalised cluster complex
of Fomin and Reading [19] to the “M-triangle” of Armstrong’s generalised non-crossing
partitions. The F -triangle is a certain refined face count in the generalised cluster
complex. We do not give the definition here and, instead, refer the reader to [1, 27],
because it will not be important in what follows. It suffices to know that, again fixing
a finite root system Φ of rank n and a positive integer m, the F -triangle FΦm (x, y) for
the generalised cluster complex ∆m (Φ) is a polynomial in x and y, and that it was
computed in [27] for all types. What we need here is that it was shown in [27, Sec. 11,
DECOMPOSITION NUMBERS FOR FINITE COXETER GROUPS
55
Prop. D] that
(1 − xy)
n
x(1 + y) xy
,
1 − xy 1 − xy
X
n − 1 m(n − 1) m(n − 1) + s − r − 1
s r
=
xy 2
s−r
r
s−1
r,s≥0
n − 2 m(n − 1) m(n − 1) + s − r − 1
+
s−r
r
s
n − 1 m(n − 1) − 1 m(n − 1) + s − r − 1
+m
s−r
r−2
s−1
!
n − 1 m(n − 1) m(n − 1) + s − r − 2
. (9.1)
−m
s−r−2
r
s−1
FDmn
The “M-triangle” of NC m (Φ) is the polynomial defined by
X
MΦm (x, y) =
µ(u, w) xrk u y rk w ,
u,w∈N C m (Φ)
where µ(u, w) is the Möbius function in NC m (Φ). It is called “triangle” because the
Möbius function µ(u, w) vanishes unless u ≤ w, and, thus, the only coefficients in the
polynomial which may be non-zero are the coefficients of xk y l with 0 ≤ k ≤ l ≤ n.
An equivalent object is the dual M-triangle, which is defined by
X
∗
∗
(MΦm )∗ (x, y) =
µ∗ (u, w) xrk w y rk u ,
u,w∈(N C m (Φ))∗
where (NC m (Φ))∗ denotes the poset dual to NC m (Φ) (i.e., the poset which arises from
NC m (Φ) by reversing all order relations), where µ∗ denotes the Möbius function in
(NC m (Φ))∗ , and where rk∗ denotes the rank function in (NC m (Φ))∗ . It is equivalent
since, obviously, we have
(MΦm )∗ (x, y) = (xy)n MΦm (1/x, 1/y).
(9.2)
Given this notation, Armstrong’s F = M (Ex-)Conjecture [1, Conjecture 5.3.2] reads
as follows.
Conjecture FM. For any finite root system Φ of rank n, we have
1+y y−x
m
n
m
FΦ (x, y) = y MΦ
.
,
y−x
y
Equivalently,
(1 − xy)
n
FΦm
x(1 + y) xy
,
1 − xy 1 − xy
=
X
∗
µ∗ (u, w) (−x)rk
w
∗
(−y)rk u .
(9.3)
u,w∈(N C m (Φ))∗
So, Equation (9.1) provides an expression for the left-hand side of (9.3) for Φ = Dn .
With our result on rank-selected chain enumeration in NC m (Dn ) given in Corollary 19,
we are now able to calculate the right-hand side of (9.3) directly. As we mentioned
already in the Introduction, together with the results from [27, 28], this completes a
56
C. KRATTENTHALER AND T. W. MÜLLER
computational case-by-case proof of Conjecture FM. A case-free proof had been found
earlier by Tzanaki in [38].
The only ingredient that we need for the proof is the well-known link between chain
enumeration and the Möbius function. (The reader should consult [33, Sec. 3.11] for
more information on this topic.) Given a poset P and two elements u and w, u ≤ w,
in the poset, the zeta polynomial of the interval [u, w], denoted by Z(u, w; z), is the
number of (multi)chains from u to w of length z. (It can be shown that this is indeed
a polynomial in z.) Then the Möbius function of u and w is equal to µ(u, w) =
Z(u, w; −1).
Proof of Conjecture FM in type Dn . We now compute the right-hand side of (9.3), that
is,
X
∗
∗
µ∗ (u, w)(−x)rk w (−y)rk u .
u,w∈(N C m (Dn ))∗
In order to compute the coefficient of xs y r in this expression,
X
µ∗ (u, w),
(−1)r+s
u,w∈(N C m (Dn ))∗
with rk∗ u=r and rk∗ w=s
we compute the sum of all corresponding zeta polynomials (in the variable z), multiplied
by (−1)r+s ,
X
Z(u, w; z),
(−1)r+s
u,w∈(N C m (Dn ))∗
with rk∗ u=r and rk∗ w=s
and then put z = −1.
For computing this sum of zeta polynomials, we must set l = z + 2, n−s1 = s, sl = r,
s2 + s3 + · · · + sl−1 = s − r in (8.40), and then sum the resulting expression over all
possible s2 , s3 , . . . , sl−1 . (The reader should keep in mind that the roles of s1 , s2 , . . . , sl
in Corollary 19 have to be reversed, since we are aiming at computing zeta polynomials
in the poset dual to NC m (Dn ).) By using the Chu–Vandermonde summation, one
obtains
m(n − 1) − 1 zm(n − 1) n − 1
m(n − 1) zm(n − 1) n − 1
+m
2
s−1
s−r
r−2
s−1
s−r
r
m(n − 1) zm(n − 1) n − 2
m(n − 1) zm(n − 1) − 1 n − 1
.
+
+ zm
s
s−r
r
s−1
s−r−2
r
If we put z = −1 in this expression and multiply it by (−1)r+s , then we obtain exactly
the coefficient of xs y r in (9.1).
10. A conjecture of Armstrong on maximal intervals containing a
random multichain
Given a finite root system of rank n, Conjecture 3.5.13 in [1] says the following:
If we choose an l-multichain uniformly at random from the set
π1 ≤ π2 ≤ · · · ≤ πl : πi ∈ NC m (Φ), i = 1, . . . , l, and rk(π1 ) = i ,
(10.1)
DECOMPOSITION NUMBERS FOR FINITE COXETER GROUPS
57
then the expected number of maximal intervals in NC m (Φ) containing this multichain
is
Narm (Φ, n − i)
,
(10.2)
Nar1 (Φ, n − i)
where Narm (Φ, i) is the i-th Fuß–Narayana number associated to NC m (Φ), that is,
the number of elements of NC m (Φ) of rank i. In particular, this expected value is
independent of l.
We show in this section that, for types An and Bn , the conjecture follows easily from
Edelman’s (8.14) respectively Armstrong’s (8.25) (presumably, this fact constituted the
evidence for setting up the conjecture), while an analogous computation using our new
result (8.40) demonstrates that it fails for type Dn . At the end of this section, we
comment on what we think happens for the exceptional types.
The computation of the expected value in the above conjecture can be approached in
the following way. One first observes that a maximal interval in NC m (Φ) is an interval
between an element π0 of rank 0 and the global maximum (c; ε, . . . , ε). Therefore, to
compute the proposed expected value, we may count the number of chains
π0 ≤ π1 ≤ π2 ≤ · · · ≤ πl ,
rk(π0 ) = 0 and rk(π1 ) = i,
(10.3)
and divide this number by the total number of all chains in (10.1). Clearly, in types
An , Bn , and Dn , this kind of chain enumeration can be easily accessed by (8.14), (8.25),
and (8.40), respectively.
We begin with type An . By (8.14), the number of chains (10.3) equals
X
1
m(n + 1)
n + 1 m(n + 1) m(n + 1)
···
sl+1
s2
i
0
n+1
s2 +···+sl+1 =n−i
1
m(n + 1) ml(n + 1)
=
,
n−i
i
n+1
while the number of chains in (10.1) equals
X
1
m(n + 1)
n + 1 m(n + 1)
···
sl+1
s2
i
n+1
s +···+s
=n−i
2
l+1
1
n + 1 ml(n + 1)
=
.
n−i
i
n+1
In both cases, we used the multivariate Chu–Vandermonde summation to evaluate the
sums over s2 , . . . , sl+1 . The quotient of the two numbers is
1
m(n + 1)
n + 1 m(n + 1)
1
i
i
n+1
n+1 n−i
,
=
n+1
n+1 n+1
1
1
i
i
n+1
n+1 n−i
which by (8.14) with n replaced by n + 1, l = 2, s1 = n − i, and s2 = i agrees indeed
with (10.2) for Φ = An .
58
C. KRATTENTHALER AND T. W. MÜLLER
For type Bn , there is an analogous computation using (8.25), the details of which we
leave to the reader. The result is that the desired expected value equals
mn
n
mn
i
n−i
i
,
= n
n
n
i
i
n−i
which by (8.25) with l = 2, s1 = n − i, and s2 = i agrees indeed with (10.2) for Φ = Bn .
The analogous computation for type Dn uses (8.40). The number of chains (10.3)
equals
X
m(n − 1)
n − 1 m(n − 1) m(n − 1)
···
2
sl+1
s
i
0
2
s2 +···+sl+1 =n−i
X
m(n − 1)
n − 1 m(n − 1) − 1 m(n − 1)
···
+m
sl+1
s
i
−
2
0
2
s +···+s
=n−i
2
l+1
n − 1 m(n − 1) m(n − 1)
+m
s2
i
0
j=3 s2 +···+sl+1 =n−i
m(n − 1)
m(n − 1) − 1
···
···
sl+1
sj − 2
m(n − 1) − 1 ml(n − 1)
m(n − 1) ml(n − 1)
+m
=2
n−i
i−2
n−i
i
m(n − 1) ml(n − 1) − 1
,
(10.4)
+ m(l − 1)
n−i−2
i
l+1
X
X
while the number of chains in (10.1) equals
X
m(n − 1)
n − 1 m(n − 1)
···
2
sl+1
s
i
2
s +···+s
=n−i
2
l+1
m(n − 1)
m(n − 1) − 1
n − 1 m(n − 1)
···
···
+m
sl+1
s
−
2
s
i
j
2
j=2 s2 +···+sl+1 =n−i
X
m(n − 1)
n − 2 m(n − 1)
···
+
sl+1
s
i
−
2
2
s2 +···+sl+1 =n−i
n − 2 ml(n − 1)
n − 1 ml(n − 1) − 1
n − 1 ml(n − 1)
.
+
+ ml
=2
n−i
i−2
n−i−2
i
n−i
i
(10.5)
l+1
X
X
The quotient of (10.4) and (10.5) gives the desired expected value. It is, however, not
independent of l, and therefore Armstrong’s conjecture does not hold for Φ = Dn .
In the case that Φ is of exceptional type, then, as we outline in the next section, the
knowledge of the corresponding decomposition numbers (see the Appendix) allows one
to access the rank selected chain enumeration. Using this, the approach for computing
the expected value proposed by Armstrong that we used above for the classical types
DECOMPOSITION NUMBERS FOR FINITE COXETER GROUPS
59
can be carried through as well for the exceptional types. We have not done this, but we
expect that, similarly to the case of Dn , for most exceptional types the expected value
will depend on l, so that Armstrong’s conjecture will probably also fail in these cases.
11. Chain enumeration in the poset of generalised non-crossing
partitions for the exceptional types
Although it is not the main topic of our paper, we want to briefly demonstrate
in this section that the knowledge of the decomposition numbers also enables one to
do refined enumeration in the generalised non-crossing partition posets NC m (Φ) for
exceptional root systems Φ (of rank n). We restrict the following considerations to the
rank-selected chain enumeration. This means that we want to count the number of all
multi-chains π1 ≤ π2 ≤ · · · ≤ πl−1 in NC m (Φ), where πi is of rank s1 + s2 + · · · + si ,
i = 1, 2, . . . , l − 1. Let us denote this number by RΦ (s1 , s2 , . . . , sl ), with sl = n − s1 −
s2 − · · · − sl . Now, the considerations at the beginning of the proof of Corollary 12,
leading to the factorisation (8.12) with rank constraints on the factors, are also valid
for NC m (Φ) instead of NC m (An−1 ), that is, they are independent of the underlying
root system. Hence, to determine the number RΦ (s1 , s2 , . . . , sl ), we have to count all
possible factorisations
(1)
(2) (3)
(l) (2) (3)
(l) (3)
(l)
c = w0 u1 u1 · · · u1 u2 u2 · · · u2 · · · u(2)
m um · · · um ,
(1)
under the rank constraints (8.11) and ℓT (w0 ) = s1 , where c is a Coxeter element in
W (Φ). As we remarked in the proof of Corollary 12, equivalently we may count all
factorisations
(3) (3)
(1)
(2) (2)
(l) (l)
(l)
c = w0 u1 u2 · · · u(2)
u1 u2 · · · u(3)
(11.1)
m
m · · · u1 u2 · · · um
(1)
which satisfy (8.11) and ℓT (w0 ) = s1 . We can now obtain an explicit expression by
(1)
(j)
fixing first the types of w0 and all the ui ’s. Under these constraints, the number of
factorisations (11.1) is just the corresponding decomposition number. Subsequently, we
sum the resulting expressions over all possible types.
Before we are able to state the formula which we obtain in this way, we need to recall
some standard integer partition notation (cf. e.g. [34, Sec. 7.2]). An integer partition λ
(with n parts) is an n-tuple λ = (λ1 , λ2 , . . . , λn ) of integers satisfying λ1 ≥ λ2 ≥ · · · ≥
λn ≥ 0. It is called an integer partition of N, written in symbolic notation as λ ⊢ N,
if λ1 + λ2 + · · · + λn = N. The number of parts (components) of λ of size i is denoted
by mi (λ).
Then, making again use of the notation for the multinomial coefficient introduced in
Lemma 4, the expression for RΦ (s1 , s2 , . . . , sl ) which we obtain in the way described
above is
l Y
X
m
′ N (T (1) , T (2) , T (2) , . . . , T (l) )
,
(11.2)
Φ 0
1
2
n
m1 (λ(j) ), m2 (λ(j) ), . . . , mn (λ(j) )
j=2
P′
where
is taken over all integer partitions λ(2) , λ(3) , . . . , λ(l) satisfying λ(2) ⊢ s2 ,
(1)
(1)
(j)
λ(3) ⊢ s3 , . . . , λ(l) ⊢ sl , over all types T0 with rk(T0 ) = s1 , and over all types Ti
(j)
(j)
with rk(Ti ) = λi , i = 1, 2, . . . , n, j = 2, 3, . . . , l.
By way of example, using this formula and the values of the decomposition numbers NE8 (. . . ) given in Appendix A.7 (and a computer), we obtain that the number
60
C. KRATTENTHALER AND T. W. MÜLLER
RE8 (4, 2, 1, 1) of all chains π1 ≤ π2 ≤ π3 in NC m (E8 ), where π1 is of rank 4, π2 is of
rank 6, and π3 is of rank 7, is given by
75m3 (8055m − 1141)
2
(which, by the independence (2.2) of decomposition numbers from the order of the types,
is also equal to RE8 (4, 1, 2, 1) and RE8 (4, 1, 1, 2)), while the number RE8 (2, 4, 1, 1) of all
chains π1 ≤ π2 ≤ π3 in NC m (E8 ), where π1 is of rank 2, π2 is of rank 6, and π3 is of
rank 7, is given by
75m3 (73125m3 − 58950m2 + 15635m − 2154)
8
(which is also equal to RE8 (2, 1, 4, 1) and RE8 (2, 1, 1, 4)).
Acknowledgements
The authors thank the anonymous referee for a very careful reading of the original
manuscript.
Appendix A. The decomposition numbers for the exceptional types
A.1. The decomposition numbers for type I2 (a) [27, Sec. 13]. We have
NI2 (a) (I2 (a)) = 1, NI2 (a) (A1 , A1 ) = a, NI2 (a) (A1 ) = a, NI2 (a) (∅) = 1, all other numbers NI2 (a) (T1 , T2 , . . . , Td ) being zero.
A.2. The decomposition numbers for type H3 [27, Sec. 14]. We have NH3 (H3 ) = 1,
NH3 (A21 , A1 ) = 5, NH3 (A2 , A1 ) = 5, NH3 (I2 (5), A1 ) = 5, NH3 (A1 , A1 , A1 ) = 50, plus the
assignments implied by (2.2) and (2.3), all other numbers NH3 (T1 , T2 , . . . , Td ) being
zero.
A.3. The decomposition numbers for type H4 [27, Sec. 15]. We have NH4 (H4 ) = 1,
NH4 (A1 ∗ A2 , A1 ) = 15, NH4 (A3 , A1 ) = 15, NH4 (H3 , A1 ) = 15, NH4 (A1 ∗ I2 (5), A1 ) =
15, NH4 (A21 , A21 ) = 30, NH4 (A21 , A2 ) = 30, NH4 (A21 , I2 (5)) = 15, NH4 (A2 , A2 ) = 5,
NH4 (A2 , I2 (5)) = 15, NH4 (I2 (5), I2 (5)) = 3, NH4 (A21 , A1 , A1 ) = 225, NH4 (A2 , A1 , A1 ) =
150, NH4 (I2 (5), A1 , A1 ) = 90, NH4 (A1 , A1 , A1 , A1 ) = 1350, plus the assignments implied
by (2.2) and (2.3), all other numbers NH4 (T1 , T2 , . . . , Td ) being zero.
A.4. The decomposition numbers for type F4 [27, Sec. 16]. We have NF4 (F4 ) =
1, NF4 (A1 ∗ A2 , A1 ) = 12, NF4 (B3 , A1 ) = 12, NF4 (A21 , A21 ) = 12, NF4 (A21 , B2 ) = 12,
NF4 (A2 , A2 ) = 16, NF4 (B2 , B2 ) = 3, NF4 (A21 , A1 , A1 ) = 72, NF4 (A2 , A1 , A1 ) = 48,
NF4 (B2 , A1 , A1 ) = 36, NF4 (A1 , A1 , A1 , A1 ) = 432, plus the assignments implied by (2.2)
and (2.3), all other numbers NF4 (T1 , T2 , . . . , Td ) being zero.
DECOMPOSITION NUMBERS FOR FINITE COXETER GROUPS
61
A.5. The decomposition numbers for type E6 [27, Sec. 17]. We have NE6 (E6 ) = 1,
NE6 (A1 ∗ A22 , A1 ) = 6, NE6 (A1 ∗ A4 , A1 ) = 12, NE6 (A5 , A1 ) = 6, NE6 (D5 , A1 ) = 12,
NE6 (A21 ∗ A2 , A2 ) = 36, NE6 (A22 , A2 ) = 8, NE6 (A1 ∗ A3 , A2 ) = 24, NE6 (A4 , A2 ) = 24,
NE6 (D4 , A2 ) = 4, NE6 (A21 ∗ A2 , A21 ) = 18, NE6 (A1 ∗ A3 , A21 ) = 36, NE6 (A4 , A21 ) = 36,
NE6 (D4 , A21 ) = 18, NE6 (A31 , A31 ) = 12, NE6 (A1 ∗ A2 , A31 ) = 24, NE6 (A1 ∗ A2 , A1 ∗
A2 ) = 48, NE6 (A3 , A31 ) = 36, NE6 (A3 , A1 ∗ A2 ) = 72, NE6 (A3 , A3 ) = 27, NE6 (A21 ∗
A2 , A1 , A1 ) = 144, NE6 (A22 , A1 , A1 ) = 24, NE6 (A1 ∗A3 , A1 , A1 ) = 144, NE6 (A4 , A1 , A1 ) =
144, NE6 (D4 , A1 , A1 ) = 48, NE6 (A31 , A21 , A1 ) = 180, NE6 (A31 , A2 , A1 ) = 168, NE6 (A1 ∗
A2 , A21 , A1 ) = 360, NE6 (A1 ∗ A2 , A2 , A1 ) = 336, NE6 (A3 , A21 , A1 ) = 378, NE6 (A3 , A2 ,
A1 ) = 180, NE6 (A21 , A21 , A21 ) = 432, NE6 (A2 , A21 , A21 ) = 504, NE6 (A2 , A2 , A21 ) = 288,
NE6 (A2 , A2 , A2 ) = 160, NE6 (A21 , A21 , A1 , A1 ) = 2376, NE6 (A2 , A21 , A1 , A1 ) = 1872,
NE6 (A2 , A2 , A1 , A1 ) = 1056, NE6 (A31 , A1 , A1 , A1 ) = 864, NE6 (A1 ∗ A2 , A1 , A1 , A1 ) =
1728, NE6 (A3 , A1 , A1 , A1 ) = 1296, NE6 (A21 , A1 , A1 , A1 , A1 ) = 10368, NE6 (A2 , A1 , A1 , A1 ,
A1 ) = 6912, NE6 (A1 , A1 , A1 , A1 , A1 , A1 ) = 41472, plus the assignments implied by (2.2)
and (2.3), all other numbers NE6 (T1 , T2 , . . . , Td ) being zero.
A.6. The decomposition numbers for type E7 [28, Sec. 6]. We have NE7 (E7 ) =
1, NE7 (E6 , A1 ) = 9, NE7 (D6 , A1 ) = 9, NE7 (A6 , A1 ) = 9, NE7 (A1 ∗ D5 , A1 ) = 9,
NE7 (A1 ∗A5 , A1 ) = 9, NE7 (A2 ∗D4 , A1 ) = 0, NE7 (A2 ∗A4 , A1 ) = 9, NE7 (A21 ∗D4 , A1 ) = 0,
NE7 (A21 ∗A4 , A1 ) = 0, NE7 (A23 , A1 ) = 0, NE7 (A1 ∗A2 ∗A3 , A1 ) = 9, NE7 (A31 ∗A3 , A1 ) = 0,
NE7 (A32 , A1 ) = 0, NE7 (A21 ∗ A22 , A1 ) = 0, NE7 (A41 ∗ A2 , A1 ) = 0, NE7 (A61 , A1 ) = 0,
NE7 (D5 , A2 ) = 18, NE7 (A5 , A2 ) = 30, NE7 (A1 ∗ A4 , A2 ) = 54, NE7 (A1 ∗ D4 , A2 ) = 9,
NE7 (A2 ∗ A3 , A2 ) = 36, NE7 (A21 ∗ A3 , A2 ) = 36, NE7 (A1 ∗ A22 , A2 ) = 36, NE7 (A31 ∗
A2 , A2 ) = 12, NE7 (A51 , A2 ) = 0, NE7 (D5 , A21 ) = 54, NE7 (A5 , A21 ) = 63, NE7 (A1 ∗
D4 , A21 ) = 27, NE7 (A1 ∗ A4 , A21 ) = 81, NE7 (A2 ∗ A3 , A21 ) = 27, NE7 (A21 ∗ A3 , A21 ) = 27,
NE7 (A1 ∗ A22 , A21 ) = 27, NE7 (A31 ∗ A2 , A21 ) = 9, NE7 (A51 , A21 ) = 0, NE7 (D5 , A1 , A1 ) =
162, NE7 (A5 , A1 , A1 ) = 216, NE7 (A1 ∗ D4 , A1 , A1 ) = 81, NE7 (A1 ∗ A4 , A1 , A1 ) = 324,
NE7 (A2 ∗ A3 , A1 , A1 ) = 162, NE7 (A21 ∗ A3 , A1 , A1 ) = 162, NE7 (A1 ∗ A22 , A1 , A1 ) = 162,
NE7 (A31 ∗ A2 , A1 , A1 ) = 54, NE7 (A51 , A1 , A1 ) = 0, NE7 (D4 , A3 ) = 9, NE7 (A4 , A3 ) = 54,
NE7 (A1 ∗ A3 , A3 ) = 135, NE7 (A22 , A3 ) = 54, NE7 (A21 ∗ A2 , A3 ) = 162, NE7 (A41 , A3 ) = 27,
NE7 (D4 , A1 ∗A2 ) = 45, NE7 (A4 , A1 ∗A2 ) = 162, NE7 (A1 ∗A3 , A1 ∗A2 ) = 243, NE7 (A22 , A1 ∗
A2 ) = 54, NE7 (A21 ∗ A2 , A1 ∗ A2 ) = 162, NE7 (A41 , A1 ∗ A2 ) = 27, NE7 (D4 , A31 ) = 30,
NE7 (A4 , A31 ) = 99, NE7 (A1 ∗ A3 , A31 ) = 126, NE7 (A22 , A31 ) = 18, NE7 (A21 ∗ A2 , A31 ) = 54,
NE7 (A41 , A31 ) = 9, NE7 (D4 , A2 , A1 ) = 81, NE7 (A4 , A2 , A1 ) = 378, NE7 (A1 ∗ A3 , A2 , A1 ) =
783, NE7 (A22 , A2 , A1 ) = 270, NE7 (A21 ∗ A2 , A2 , A1 ) = 810, NE7 (A41 , A2 , A1 ) = 135,
NE7 (D4 , A21 , A1 ) = 243, NE7 (A4 , A21 , A1 ) = 891, NE7 (A1 ∗ A3 , A21 , A1 ) = 1377, NE7 (A22 ,
A21 , A1 ) = 324, NE7 (A21 ∗ A2 , A21 , A1 ) = 972, NE7 (A41 , A21 , A1 ) = 162, NE7 (D4 , A1 , A1 ,
A1 ) = 729, NE7 (A4 , A1 , A1 , A1 ) = 2916, NE7 (A1 ∗ A3 , A1 , A1 , A1 ) = 5103, NE7 (A22 , A1 ,
A1 , A1 ) = 1458, NE7 (A21 ∗ A2 , A1 , A1 , A1 ) = 4374, NE7 (A41 , A1 , A1 , A1 ) = 729, NE7 (A3 ,
A3 , A1 ) = 486, NE7 (A3 , A1 ∗ A2 , A1 ) = 1458, NE7 (A3 , A31 , A1 ) = 891, NE7 (A1 ∗ A2 , A1 ∗
A2 , A1 ) = 2430, NE7 (A1 ∗A2 , A31 , A1 ) = 1215, NE7 (A31 , A31 , A1 ) = 540, NE7 (A3 , A2 , A2 ) =
432, NE7 (A1 ∗ A2 , A2 , A2 ) = 1188, NE7 (A31 , A2 , A2 ) = 711, NE7 (A3 , A2 , A21 ) = 1053,
NE7 (A1 ∗A2 , A2 , A21 ) = 2349, NE7 (A31 , A2 , A21 ) = 1323, NE7 (A3 , A21 , A21 ) = 2430, NE7 (A1 ∗
A2 , A21 , A21 ) = 3402, NE7 (A31 , A21 , A21 ) = 1539, NE7 (A3 , A2 , A1 , A1 ) = 3402, NE7 (A1 ∗
A2 , A2 , A1 , A1 ) = 8262, NE7 (A31 , A2 , A1 , A1 ) = 4779, NE7 (A3 , A21 , A1 , A1 ) = 8019,
NE7 (A1 ∗ A2 , A21 , A1 , A1 ) = 13851, NE7 (A31 , A21 , A1 , A1 ) = 7047, NE7 (A3 , A1 , A1 , A1 ,
A1 ) = 26244, NE7 (A1 ∗ A2 , A1 , A1 , A1 , A1 ) = 52488, NE7 (A31 , A1 , A1 , A1 , A1 ) = 28431,
62
C. KRATTENTHALER AND T. W. MÜLLER
NE7 (A2 , A2 , A2 , A1 ) = 2916, NE7 (A2 , A2 , A21 , A1 ) = 6561, NE7 (A2 , A21 , A21 , A1 ) = 13122,
NE7 (A21 , A21 , A21 , A1 ) = 19683, NE7 (A2 , A2 , A1 , A1 , A1 ) = 21870, NE7 (A2 , A21 , A1 , A1 ,
A1 ) = 45927, NE7 (A21 , A21 , A1 , A1 , A1 ) = 78732, NE7 (A2 , A1 , A1 , A1 , A1 , A1 ) = 157464,
NE7 (A21 , A1 , A1 , A1 , A1 , A1 ) = 295245, NE7 (A1 , A1 , A1 , A1 , A1 , A1 , A1 ) = 1062882, plus
the assignments implied by (2.2) and (2.3), all other numbers NE7 (T1 , T2 , . . . , Td ) being
zero.
A.7. The decomposition numbers for type E8 [28, Sec. 7]. We have NE8 (E8 ) = 1,
NE8 (E7 , A1 ) = 15, NE8 (D7 , A1 ) = 15, NE8 (A7 , A1 ) = 15, NE8 (A1 ∗ E6 , A1 ) = 15,
NE8 (A1 ∗D6 , A1 ) = 0, NE8 (A1 ∗A6 , A1 ) = 15, NE8 (A2 ∗D5 , A1 ) = 15, NE8 (A2 ∗A5 , A1 ) =
0, NE8 (A21 ∗D5 , A1 ) = 0, NE8 (A21 ∗A5 , A1 ) = 0, NE8 (A3 ∗D4 , A1 ) = 0, NE8 (A3 ∗A4 , A1 ) =
15, NE8 (A1 ∗A2 ∗D4 , A1 ) = 0, NE8 (A1 ∗A2 ∗A4 , A1 ) = 15, NE8 (A31 ∗D4 , A1 ) = 0, NE8 (A31 ∗
A4 , A1 ) = 0, NE8 (A1 ∗ A23 , A1 ) = 0, NE8 (A22 ∗ A3 , A1 ) = 0, NE8 (A21 ∗ A2 ∗ A3 , A1 ) = 0,
NE8 (A41 ∗A3 , A1 ) = 0, NE8 (A1 ∗A32 , A1 ) = 0, NE8 (A31 ∗A22 , A1 ) = 0, NE8 (A51 ∗A2 , A1 ) = 0,
NE8 (A71 , A1 ) = 0, NE8 (E6 , A2 ) = 20, NE8 (D6 , A2 ) = 15, NE8 (A6 , A2 ) = 60, NE8 (A1 ∗
D5 , A2 ) = 60, NE8 (A1 ∗ A5 , A2 ) = 60, NE8 (A2 ∗ D4 , A2 ) = 20, NE8 (A2 ∗ A4 , A2 ) = 90,
NE8 (A23 , A2 ) = 45, NE8 (A21 ∗D4 , A2 ) = 0, NE8 (A21 ∗A4 , A2 ) = 90, NE8 (A1 ∗A2 ∗A3 , A2 ) =
90, NE8 (A31 ∗A3 , A2 ) = 0, NE8 (A32 , A2 ) = 0, NE8 (A21 ∗A22 , A2 ) = 45, NE8 (A41 ∗A2 , A2 ) = 0,
NE8 (A61 , A2 ) = 0, NE8 (E6 , A21 ) = 45, NE8 (D6 , A21 ) = 90, NE8 (A6 , A21 ) = 135, NE8 (A1 ∗
D5 , A21 ) = 135, NE8 (A1 ∗ A5 , A21 ) = 135, NE8 (A2 ∗ D4 , A21 ) = 45, NE8 (A2 ∗ A4 , A21 ) = 90,
NE8 (A23 , A21 ) = 45, NE8 (A21 ∗ D4 , A21 ) = 0, NE8 (A21 ∗ A4 , A21 ) = 90, NE8 (A1 ∗ A2 ∗
A3 , A21 ) = 90, NE8 (A31 ∗ A3 , A21 ) = 0, NE8 (A32 , A21 ) = 0, NE8 (A21 ∗ A22 , A21 ) = 45,
NE8 (A41 ∗ A2 , A21 ) = 0, NE8 (A61 , A21 ) = 0, NE8 (E6 , A1 , A1 ) = 150, NE8 (D6 , A1 , A1 ) = 225,
NE8 (A6 , A1 , A1 ) = 450, NE8 (A1 ∗ D5 , A1 , A1 ) = 450, NE8 (A1 ∗ A5 , A1 , A1 ) = 450,
NE8 (A2 ∗ D4 , A1 , A1 ) = 150, NE8 (A2 ∗ A4 , A1 , A1 ) = 450, NE8 (A23 , A1 , A1 ) = 225,
NE8 (A21 ∗ D4 , A1 , A1 ) = 0, NE8 (A21 ∗ A4 , A1 , A1 ) = 450, NE8 (A1 ∗ A2 ∗ A3 , A1 , A1 ) =
450, NE8 (A31 ∗ A3 , A1 , A1 ) = 0, NE8 (A32 , A1 , A1 ) = 0, NE8 (A21 ∗ A22 , A1 , A1 ) = 225,
NE8 (A41 ∗ A2 , A1 , A1 ) = 0, NE8 (A61 , A1 , A1 ) = 0, NE8 (D5 , A3 ) = 45, NE8 (A5 , A3 ) = 90,
NE8 (A1 ∗ A4 , A3 ) = 315, NE8 (A1 ∗ D4 , A3 ) = 45, NE8 (A2 ∗ A3 , A3 ) = 270, NE8 (A21 ∗
A3 , A3 ) = 270, NE8 (A1 ∗ A22 , A3 ) = 225, NE8 (A31 ∗ A2 , A3 ) = 225, NE8 (A51 , A3 ) = 0,
NE8 (D5 , A1 ∗ A2 ) = 195, NE8 (A5 , A1 ∗ A2 ) = 390, NE8 (A1 ∗ A4 , A1 ∗ A2 ) = 690, NE8 (A1 ∗
D4 , A1 ∗ A2 ) = 195, NE8 (A2 ∗ A3 , A1 ∗ A2 ) = 495, NE8 (A21 ∗ A3 , A1 ∗ A2 ) = 495, NE8 (A1 ∗
A22 , A1 ∗ A2 ) = 300, NE8 (A31 ∗ A2 , A1 ∗ A2 ) = 300, NE8 (A51 , A1 ∗ A2 ) = 0, NE8 (D5 , A31 ) =
150, NE8 (A5 , A31 ) = 300, NE8 (A1 ∗ A4 , A31 ) = 375, NE8 (A1 ∗ D4 , A31 ) = 150, NE8 (A2 ∗
A3 , A31 ) = 225, NE8 (A21 ∗ A3 , A31 ) = 225, NE8 (A1 ∗ A22 , A31 ) = 75, NE8 (A31 ∗ A2 , A31 ) = 75,
NE8 (A51 , A31 ) = 0, NE8 (D5 , A2 , A1 ) = 375, NE8 (A5 , A2 , A1 ) = 750, NE8 (A1 ∗A4 , A2 , A1 ) =
1950, NE8 (A1 ∗ D4 , A2 , A1 ) = 375, NE8 (A2 ∗ A3 , A2 , A1 ) = 1575, NE8 (A21 ∗ A3 , A2 , A1 ) =
1575, NE8 (A1 ∗ A22 , A2 , A1 ) = 1200, NE8 (A31 ∗ A2 , A2 , A1 ) = 1200, NE8 (A51 , A2 , A1 ) =
0, NE8 (D5 , A21 , A1 ) = 1125, NE8 (A5 , A21 , A1 ) = 2250, NE8 (A1 ∗ A4 , A21 , A1 ) = 3825,
NE8 (A1 ∗ D4 , A21 , A1 ) = 1125, NE8 (A2 ∗ A3 , A21 , A1 ) = 2700, NE8 (A21 ∗ A3 , A21 , A1 ) = 2700,
NE8 (A1 ∗ A22 , A21 , A1 ) = 1575, NE8 (A31 ∗ A2 , A21 , A1 ) = 1575, NE8 (A51 , A21 , A1 ) = 0,
NE8 (D5 , A1 , A1 , A1 ) = 3375, NE8 (A5 , A1 , A1 , A1 ) = 6750, NE8 (A1 ∗ A4 , A1 , A1 , A1 ) =
13500, NE8 (A1 ∗ D4 , A1 , A1 , A1 ) = 3375, NE8 (A2 ∗ A3 , A1 , A1 , A1 ) = 10125, NE8 (A21 ∗
A3 , A1 , A1 , A1 ) = 10125, NE8 (A1 ∗ A22 , A1 , A1 , A1 ) = 6750, NE8 (A31 ∗ A2 , A1 , A1 , A1 ) =
6750, NE8 (A51 , A1 , A1 , A1 ) = 0, NE8 (D4 , D4 ) = 5, NE8 (D4 , A4 ) = 15, NE8 (A4 , A4 ) = 138,
NE8 (D4 , A1 ∗ A3 ) = 105, NE8 (A4 , A1 ∗ A3 ) = 390, NE8 (A1 ∗ A3 , A1 ∗ A3 ) = 1155,
NE8 (D4 , A22 ) = 35, NE8 (A4 , A22 ) = 180, NE8 (A1 ∗ A3 , A22 ) = 360, NE8 (A22 , A22 ) = 95,
DECOMPOSITION NUMBERS FOR FINITE COXETER GROUPS
63
NE8 (D4 , A21 ∗ A2 ) = 135, NE8 (A4 , A21 ∗ A2 ) = 630, NE8 (A1 ∗ A3 , A21 ∗ A2 ) = 1035,
NE8 (A22 , A21 ∗ A2 ) = 270, NE8 (A21 ∗ A2 , A21 ∗ A2 ) = 495, NE8 (D4 , A41 ) = 30, NE8 (A4 , A41 ) =
165, NE8 (A1 ∗ A3 , A41 ) = 255, NE8 (A22 , A41 ) = 60, NE8 (A21 ∗ A2 , A41 ) = 135, NE8 (A41 , A41 ) =
30, NE8 (D4 , A3 , A1 ) = 225, NE8 (A4 , A3 , A1 ) = 1215, NE8 (A1 ∗ A3 , A3 , A1 ) = 4050,
NE8 (A22 , A3 , A1 ) = 1575, NE8 (A21 ∗A2 , A3 , A1 ) = 5400, NE8 (A41 , A3 , A1 ) = 1350, NE8 (D4 ,
A1 ∗ A2 , A1 ) = 975, NE8 (A4 , A1 ∗ A2 , A1 ) = 4590, NE8 (A1 ∗ A3 , A1 ∗ A2 , A1 ) = 10800,
NE8 (A22 , A1 ∗ A2 , A1 ) = 3450, NE8 (A21 ∗ A2 , A1 ∗ A2 , A1 ) = 9900, NE8 (A41 , A1 ∗ A2 , A1 ) =
2475, NE8 (D4 , A31 , A1 ) = 750, NE8 (A4 , A31 , A1 ) = 3375, NE8 (A1 ∗ A3 , A31 , A1 ) = 6750,
NE8 (A22 , A31 , A1 ) = 1875, NE8 (A21 ∗A2 , A31 , A1 ) = 4500, NE8 (A41 , A31 , A1 ) = 1125, NE8 (D4 ,
A2 , A2 ) = 175, NE8 (A4 , A2 , A2 ) = 1140, NE8 (A1 ∗A3 , A2 , A2 ) = 3300, NE8 (A22 , A2 , A2 ) =
1300, NE8 (A21 ∗ A2 , A2 , A2 ) = 4500, NE8 (A41 , A2 , A2 ) = 1125, NE8 (D4 , A2 , A21 ) = 675,
NE8 (A4 , A2 , A21 ) = 3015, NE8 (A1 ∗A3 , A2 , A21 ) = 8550, NE8 (A22 , A2 , A21 ) = 2925, NE8 (A21 ∗
A2 , A2 , A21 ) = 9000, NE8 (A41 , A2 , A21 ) = 2250, NE8 (D4 , A21 , A21 ) = 1800, NE8 (A4 , A21 ,
A21 ) = 8640, NE8 (A1 ∗ A3 , A21 , A21 ) = 17550, NE8 (A22 , A21 , A21 ) = 5175, NE8 (A21 ∗ A2 , A21 ,
A21 ) = 13500, NE8 (A41 , A21 , A21 ) = 3375, NE8 (D4 , A2 , A1 , A1 ) = 1875, NE8 (A4 , A2 , A1 ,
A1 ) = 9450, NE8 (A1 ∗ A3 , A2 , A1 , A1 ) = 27000, NE8 (A22 , A2 , A1 , A1 ) = 9750, NE8 (A21 ∗
A2 , A2 , A1 , A1 ) = 31500, NE8 (A41 , A2 , A1 , A1 ) = 7875, NE8 (D4 , A21 , A1 , A1 ) = 5625,
NE8 (A4 , A21 , A1 , A1 ) = 26325, NE8 (A1 ∗ A3 , A21 , A1 , A1 ) = 60750, NE8 (A22 , A21 , A1 , A1 ) =
19125, NE8 (A21 ∗ A2 , A21 , A1 , A1 ) = 54000, NE8 (A41 , A21 , A1 , A1 ) = 13500, NE8 (D4 , A1 , A1 ,
A1 , A1 ) = 16875, NE8 (A4 , A1 , A1 , A1 , A1 ) = 81000, NE8 (A1 ∗ A3 , A1 , A1 , A1 , A1 ) =
202500, NE8 (A22 , A1 , A1 , A1 , A1 ) = 67500, NE8 (A21 ∗ A2 , A1 , A1 , A1 , A1 ) = 202500,
NE8 (A41 , A1 , A1 , A1 , A1 ) = 50625, NE8 (A3 , A3 , A2 ) = 1350, NE8 (A3 , A1 ∗ A2 , A2 ) = 5175,
NE8 (A3 , A31 , A2 ) = 3825, NE8 (A1 ∗ A2 , A1 ∗ A2 , A2 ) = 15000, NE8 (A1 ∗ A2 , A31 , A2 ) =
9825, NE8 (A31 , A31 , A2 ) = 6000, NE8 (A3 , A3 , A21 ) = 4050, NE8 (A3 , A1 ∗ A2 , A21 ) = 13500,
NE8 (A3 , A31 , A21 ) = 9450, NE8 (A1 ∗ A2 , A1 ∗ A2 , A21 ) = 30825, NE8 (A1 ∗ A2 , A31 , A21 ) =
17325, NE8 (A31 , A31 , A21 ) = 7875, NE8 (A3 , A3 , A1 , A1 ) = 12150, NE8 (A3 , A1 ∗A2 , A1 , A1 ) =
42525, NE8 (A3 , A31 , A1 , A1 ) = 30375, NE8 (A1 ∗ A2 , A1 ∗ A2 , A1 , A1 ) = 106650, NE8 (A1 ∗
A2 , A31 , A1 , A1 ) = 64125, NE8 (A31 , A31 , A1 , A1 ) = 33750, NE8 (A3 , A2 , A2 , A1 ) = 10575,
NE8 (A3 , A2 , A21 , A1 ) = 29700, NE8 (A3 , A21 , A21 , A1 ) = 76950, NE8 (A1 ∗ A2 , A2 , A2 , A1 ) =
35700, NE8 (A1 ∗A2 , A2 , A21 , A1 ) = 84825, NE8 (A1 ∗A2 , A21 , A21 , A1 ) = 171450, NE8 (A31 , A2 ,
A2 , A1 ) = 25125, NE8 (A31 , A2 , A21 , A1 ) = 55125, NE8 (A31 , A21 , A21 , A1 ) = 94500, NE8 (A3 ,
A2 , A1 , A1 , A1 ) = 91125, NE8 (A3 , A21 , A1 , A1 , A1 ) = 243000, NE8 (A1 ∗ A2 , A2 , A1 , A1 ,
A1 ) = 276750, NE8 (A1 ∗A2 , A21 , A1 , A1 , A1 ) = 597375, NE8 (A31 , A2 , A1 , A1 , A1 ) = 185625,
NE8 (A31 , A21 , A1 , A1 , A1 ) = 354375, NE8 (A3 , A1 , A1 , A1 , A1 , A1 ) = 759375, NE8 (A1 ∗ A2 ,
A1 , A1 , A1 , A1 , A1 ) = 2025000, NE8 (A31 , A1 , A1 , A1 , A1 , A1 ) = 1265625, NE8 (A2 , A2 , A2 ,
A2 ) = 9350, NE8 (A2 , A2 , A2 , A21 ) = 24975, NE8 (A2 , A2 , A21 , A21 ) = 64350, NE8 (A2 , A21 ,
A21 , A21 ) = 143100, NE8 (A21 , A21 , A21 , A21 ) = 261225, NE8 (A2 , A2 , A2 , A1 , A1 ) = 78000,
NE8 (A2 , A2 , A21 , A1 , A1 ) = 203625, NE8 (A2 , A21 , A21 , A1 , A1 ) = 479250, NE8 (A21 , A21 , A21 ,
A1 , A1 ) = 951750, NE8 (A2 , A2 , A1 , A1 , A1 , A1 ) = 641250, NE8 (A2 , A21 , A1 , A1 , A1 , A1 ) =
1569375, NE8 (A21 , A21 , A1 , A1 , A1 , A1 ) = 3341250, NE8 (A2 , A1 , A1 , A1 , A1 , A1 , A1 ) =
5062500, NE8 (A21 , A1 , A1 , A1 , A1 , A1 , A1 ) = 11390625, NE8 (A1 , A1 , A1 , A1 , A1 , A1 , A1 ,
A1 ) = 37968750, plus the assignments implied by (2.2) and (2.3), all other numbers
NE8 (T1 , T2 , . . . , Td ) being zero.
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Fakultät für Mathematik, Universität Wien, Nordbergstraße 15, A-1090 Vienna,
Austria. WWW: http://www.mat.univie.ac.at/~kratt.
School of Mathematical Sciences, Queen Mary & Westfield College, University of
London, Mile End Road, London E1 4NS, United Kingdom.
WWW: http://www.maths.qmw.ac.uk/~twm/.